I'd like to point out that both Rhee and Kim were notable figures during the Independence Movement of Korea during the Japanese annexation(1910~1945).
RHEE:
Rhee was known for his very fluent English skills. He went to Baejae Hakdang(배재학당,培材學堂), notable academy back in the time which was established by Henry Appenzeller, a Methodist missionary from the United States. During his time in that academy he and other students created the Hyeopseonghoe(협성회,協成會), which is a Patriotic Enlightenment Movement group in the academy that does American styled debates on discussing the path of modernization of Korea and saving the nation. He graduated this academy in 1897 with high grades.
After his graduation he joined the Independence Club(독립협회,獨立協會), an public enlightenment group, where he continued on his activities for the modernization and enlightenment of Korea. He was caught by the police in 1899 for criticizing Emperor Gojong(Emperor of the Korean Empire). During his stay at Hanseong Prison, he wrote the book "The Spirit of Independence", which became popular among scholars.
"After seeing how strong the West was, we should have tried to become strong by working hard daily to attain wealth, strength, and civilization. The common people would have enjoyed freedom while the nation enjoyed independence, and there would have been no dangerous situation at home while there would have been no worry over aggression from abroad." - The Spirit of Independence (English Edition), Syungman Rhee, Chapter 49
He also created a Korean-English dictionary during that time too. Government officials, especially Min Yeong-hwan(민영환,閔泳煥) who wanted deep connections with the United States, took him out of prison. After Emperor Gojong 'conceded' his English skills, he was sent as an official emissary to the United States as the representative of the Emperor. During his diplomatic activities, he even had a discussion with president Theodore Roosevelt. However, after the Japan-Korea Treaty of 1905 (Korea gives up it's diplomatic sovereignty to Japan) and the Taft-Katsura Agreement(桂・タフト協定) (Japan would colonize Korea and the US agrees with that, and the US takes the Philippines), Rhee starts to question the United States on it's position towards the Korean Empire. He ultimately stops doing diplomatic work and became a student. He later studies in places like the George Washington University and Harvard. During this time, he even got close to Woodrow Wilson, who later became president.
After accumulating knowledge, his work for Korean Independence had really started. By the time he settled in Hawai'i in 1913, Korea has become a colony of the Japanese Empire. During his stay in Hawai'i, he established the Korean Christian Church(한인기독교회,韓人基督敎會) and the took over the Korean Christian Academy(한인기독학원,韓人基督學院). He taught students the Korean language and Hanja(Chinese text). He took over the Korean society of Hawai'i and contributed to Korean nationalism by leading those people.
This made him become more recognized by leaders who lead the Korean Independence. He was selected as a representative to the Paris Peace Conference in 1919 and also was appointed as Foreign Minister of the Noryeong Provisional Government. Ultimately in 1919, when the Provisional Government of the Republic of Korea in Shanghai declared the Republic of Korea Temporary Constitution (대한민국 임시 헌법, 大韓民國臨時憲法), he was selected to become the president of the provisional government.
The provisional government wasn't a very stable place. Especially due to internal fighting between the left and the right. Rhee wanted an American-like Liberal Democracy but Prime Minister Yi Dong-hwi(이동휘) claimed there should be a socialist revolution that should come first. Rhee was a very pro-American and 'diplomacy first' guy during his entire activities in the independence movement. While many on the left believed that the Koreans can defeat the Japanese with their own power, Rhee believed that the Koreans needed to hold close ties with other nations, since Koreans fighting alone would be extremely difficult. This philosophy was pretty much proven correct as the United States dropped those two bombs and it was ultimately the Americas who made Japan give up on Korea.
His grip on power during the independence movement ultimately lead him to win the First Presidential Election of the Republic of Korea in 1948. This election was not a direct election, but an indirect one. Instead of the people of South Korea voting a president directly, the members of the National Assembly were the ones who were voting. Even though it was an election that literally elected a 'president', it was just like a parliamentary system. (This ultimately changed into a direct election right after the first presidential election). 92.3% (180 votes) of the National Assembly voted for Rhee and 6.7%(13 votes) voted for Kim Gu(김구).
This specific election isn't considered unfair and undemocratic that much. The method of an indirect election is not really an undemocratic one since it's pretty much acting as if it is a parliamentary system. The members of the National Assembly (who voted Rhee into power) were elected through a single-member district direct election, which is a system that is still used in South Korea.
However, Rhee, before becoming President, was a member of the National Assembly. During the election for the National Assembly (May of 1948), he ran at the Dongdaemun-Gap district of Seoul. His opponent was Choi Neung-jin(최능진), and a far-right group called the North-West Youth Association who was associated with Rhee, stole Choi's candidate registration records and made Choi unable to run. Due to that, Choi wasn't recognized as a legit candidate and Rhee ultimately won without an election. (since he's now the only one who registered as a candidate) And during his power in office, he abused his power by manipulating elections.
KIM:
Kim Il-sung was also an independence activist, but unlike Rhee who rose to power by his education states and strategy, Kim was an activist who did physical battles against the Japanese.
I mentioned the story about Kim Il-sung on the bottom portion of this post, but here's what it says:
By the late 30's, a lot of Korean independence armies who fought the Japanese with physical battles had disappeared due to harsher environments after the Mukden Incident and the Japanese eventually taking over all of Manchuria. Kim Il-sung, who led the Northeast Anti-Japanese United Army, decided to cross into the Korean peninsula (he was active in Southern Manchuria) and invade a place called Pocheonbo(보천보). The Battle of Pocheonbo(보천보 전투) only killed 7 Japanese officers, but it was successful. It was a small battle but a significant one since it's an actual battle that the Koreans won inside of Korea. The name Kim Il-sung spread all across the country and he became a significant communist leader during the later periods of the Korean Independence Movement. He easily got a grip on the new DPRK in 1945 due to his activities before 1945.
Here are some additional details to that: The Japanese have become even more extreme on removing small Korean military groups. Because of this, Kim Il-sung moved over to Khabarovsk(Хабаровск) in the USSR and joined the Red Army. He got special agent training by the Soviets and he became the officer of the 88th International Brigade of the Red Amy. He was a part of the Red Army until the end of the Pacific War.
During his service in the Red Army, due to the support of Soviet generals such as Naum Sorkin(Наум Соркин) and Lavrentiy Beria(Лаврентий Берия), Kim has been introduced to Stalin as the guy who should lead Communist Korea. After the Soviet liberation of Korea, the Soviets needed a guy who could follow the commands of Moscow and firmly establish a soviet presence in the far east. Due to Kim's contributions on fighting the Japanese (and winning), and also his activities inside the Red Army, he was selected by Stalin to be the guy to lead Communist Korea. In September of 1945, Stalin flew Kim over to Moscow and interviewed him. It is said that he was appointed as leader of Communist Korea at this moment.
on August 8th of 1945, the Soviet Union declared war on the Japanese empire and the Red Army's tanks eventually reached Pyongyang on the 24th, thus making the northern half of Korea secure in Soviet hands. With Soviet help, Kim rose to the chairman of the Korean Communist Party and the Provisional People's Committee for North Korea. With this secure grip on the country, the Democratic People's Republic of Korea was established in September of 1948, under Kim and the Soviet's influence.
This entire process for Kim wasn't democratic or fair at all. It was ultimately the Soviet Union that made Kim rise to the top.
First, the south. In the south (the US occupation zone), the United States Army Military Government in Korea (USAMGIK) was running the show, and oversaw the elections. Originally, Syngman Rhee wasn't the front-runner for leadership, with Kim Gu more prominent. Both Rhee and Kim were long-time Korean independence activists, and both had support. The international foundation for Korean independence was a UN resolution of November 1947: a general election would be held before 31st March, 1948, on the basis of adult suffrage and secret ballots, to choose representatives who would establish "a National Government of Korea". A UN commission was formed to implement this resolution, and began operating in the south. The Soviets and the Korean leadership in their zone refused to let the UN commission operate in the Soviet occupation zone. Instead, they moved towards setting up the future people's republic - the future North Korea. This left the UN and USAMGIK unable to hold the planned elections throughout Korea. Instead, they moved towards separate elections for the south, and a divided Korea.
Kim Gu refused to accept a divided Korea. Rhee, on the other hand, was willing to accept it, temporarily (while planning reunification by force as soon as possible). Rhee, American-educated, Christian, with an Austrian wife, anti-communist, ticked all the boxes for the Americans to be willing to work with him. USAMGIK had their future president of South Korea. The elections were held, opposed and boycotted by Kim Gu. Parliament (the National Assembly) was thus elected, in May 1948. Next, a constitution. Then, presidential elections. The president was to be elected by parliament, rather than by direct election by the people.
Syngman Rhee and Kim Gu were both nominated by parliament. Kim Gu refused the nomination. There was one other candidate, An Jae-hong. An had been the head of the interim government formed by USAMGIK in 1947, as civil administrator. Like Kim Gu, An opposed a separate South Korea, and supported working with the left. Essentially, Syngman Rhee was the only candidate with a chance. Of the 198 members of the National Assembly, 196 voted. 180 votes were for Rhee, 13 for Kim (despite his refusal of the nomination), 2 for An, and 1 invalid vote for a non-candidate.
The election of the National Assembly was a fair and democratic election. Given the constitution, the election of the president (Rhee) was also fair and democratic. Rhee's opponents in the election were not to trouble him for long. Kim was assassinated the following year, in 1949, and An went to the north during the Korean War (whether voluntarily or by force isn't known). Realising that he wouldn't be re-elected by parliament in the next election, in 1952 (being too corrupt and repressive for parliament to accept), Rhee had his opponents in the National Assembly arrested, and pushed constitutional changes through the depleted and more pro-Rhee parliament. These changes made the president directly elected by the people. Rhee was re-elected with 72% of the votes. This election was during the Korean War, which probably helped Rhee. As the 1956 elections approached, Rhee pushed through more constitutional changes, to allow more than 2 terms as president. Two opponents ran against Rhee in 1956. One, Sin Ik-hui, appeared to have a good chance of victory but died, of illness, shortly before the election (there were rumours of assassination by poisoning). Sin was still on the ballot, and many (20%) voted for him. Rhee gained 55% of the votes, or 70% of valid votes after removing the invalid votes for Sin. His other opponent, Cho Bong-am, had the 30% of the valid votes. Three years later, he was executed. In the next election, Rhee's only opponent died before the election, and Rhee was unopposed. The election of the vice-president, however, was overtly and grossly rigged, triggering major protests which led to Rhee's overthrow/resignation.
Next, the north. The Korean response to independence was the formation of People's Committees for local government, and the Committee for the Preparation of Korean Independence (CPKI) in Seoul established the People's Republic of Korea (PRK), and worked as a provisional government, intending to write a constitution, hold elections, etc.. In the south, the US occupation forces outlawed the PRK and the People's Committees - altogether too hard to control and too potentially communist (leading to widespread protests, suppressed by force). Instead, USAMGIK would run the south. In the north, the Soviet occupation administration let the People's Committees run things. The Soviets also found the People's Committees too hard to control, and not communist enough. Rather than banning them, they preferred the slower but surer process of purging them and ensuring that the People's Committees were communist-dominated.
The situation in the north was not unlike that in the south: the future leader was not the leading candidate for leadership, until the leading candidate opposed the formation of separate governments in the north and south - a divided Korea. In the north, the original leading candidate was Cho Mansik. Cho opposed the division of Korea, and was no longer acceptable as leader. Cho was not a communist, so perhaps his days in power would have come to an end soon, in any case. This left Kim Il-sung as the new front-runner. Increasing communist control of the People's Committees and Soviet support led to Kim becoming the chairman of the North Korean Provisional People's Committee - the one People's Committee to rule them all, a provisional government for North Korea. In February 1948, as the UN and USAMGIK were progressing towards elections in the south, the NKPPC became the North Korean People's Committee, still with Kim in charge. The NKPC then established the DPRK in September 1948, again still with Kim in charge.
Further reading:
Armstrong, Charles K., The North Korean Revolution, 1945-1950, Cornell University Press, 2013.
Kim Bong-jin, "Paramilitary Politics under the USAMGIK and the Establishment of the ROK", Korea Journal 43(2), 289-322 (2003) https://www.ekoreajournal.net/issue/view_pop.htm?Idx=3242
u/Cenodoxus in https://www.reddit.com/r/AskHistorians/comments/15v4ws/what_was_it_about_korea_in_history_that_made/ and https://www.reddit.com/r/PoliticalScience/comments/15x2t3/i_am_hoping_some_redditors_here_are_experts_on/ and https://old.reddit.com/r/AskHistorians/comments/19nop4/how_much_input_did_the_people_of_north_korea_have/
u/dsk_oz in https://old.reddit.com/r/AskHistorians/comments/eboxhr/the_governments_of_the_koreas_were_barely_off_the/