I have to strongly disagree with /u/SUSHIKID1 here. Korean culture and people were not mix of Chinese and Japanese Culture. About in 2000 BC, we know there were various tribes and kingdoms in Korean Peninsula and Manchuria. There were records of Chinese Scholars describing various tribes and cultures of Korean Peninsula and Manchuria. They were not "Koreans", but they had similar culture to each other due to geographical proximity and constant interactions (economical and warfare). Until 670 AD, there were various proto-Korean kingdoms. After 670 AD, Korea Peninsula had been unified by one Kingdom (Silla then Goryeo). Although there were various revival movements of previous kingdoms, none of them succeeded and various cultures in Korean Peninsula had been melded together after centuries. The resulting mix is Korean People and its Culture. If you check everyone's clans, you will realize they have various different origins. For example, one person's clan might trace back to Gaya Confederacy while some other person's clan might trace back to Goguryeo, Silla or Baekje.

It is true that China had influenced Korea significantly, but many people tend to outplay this. China was sole giant Cultural and military powerhouse in Asia and Korea was impacted by its culture as much as Japan was. Not to mention Korean Peninsula was far away from China Proper. Korean people continued their own culture in Shamanism, Buddhism and Confucianism. For example, Goryeo Ware had been considered masterpices in China, and Korean people wrote several Confucian Classics discussing and sometimes disagreeing with Chinese Scholars on Confucian principles. For Japan, Korea considered Japan as being inferior than themselves so Japanese culture had almost no impact in Korea, with exception of Imjin War which brought New World goods such as red pepper and tobacco, and modern era. In fact, Baekje influenced Japanese Culture, by bringing its architecture and Korean Buddhism to Japan. Korean Scholars nowdays visit Japanese temples such as Hōryū-ji temple to learn about Baekje Architecture.

So, I have to argue that Koreans had distinct "Korean" identity to begin with. Just as Japan already had distinct "Japanese" identity. Just because Korea is between China and Japan doesn't mean Korea had no unique culture and people of her own.

EDIT: I recommend Sourcebook of Korean Civilization Vol I and II to people who want to learn about Korea. It gives primary sources on Korean Culture and its history. Most of things I said here came from these two books.

EDIT 2: For how Chinese view Korea, I have to argue that Chinese people don't know very well of Korean History to begin with. Some People think Korea was part of China due to Joseon dynasty being tribute. However, in East Asian Diplomacy system, the nations were divided to three categories: China Proper, civilized tributaries and barbarians. In this view, there was no "equal" to China. To have trade and diplomatic relations. you must be a tribute. Otherwise, you are barbarian. Japan, for example, did not have formal relations with China; however, to have trade, Japan used Satsuma Clan, which was both vassal to China and Japan, as the middle man. Korea, which borders China, did not have that luxury. Even ignoring this fact, many Korean Kingdoms did declare war against China except Joseon Dynasty, either for defense or for land grabs. Nobody disputes whether China had significant impact in Korea or not; however, to say that Korean Culture is offshoot of Chinese Culture is mere ignorance.

Linguistic analysis has shown that the Korean language is phylogenetically distinct from both the Sino-Tibetan language family (which includes the Chinese languages) and the Japonic family (which includes Japanese). This essentially means that as far as we can tell, the Korean language does not have any common ancestor with a Chinese or a Japanese language. This allows us to conjecture that Koreans started off as an independent culture/people when they first settled Korea, even though, as stated by others, a lot of what it is today has been severely influenced by both China and Japan.

Several answers in this thread mention supposed Japanese influences on Korean culture without offering evidence in support. Could any of these people highlight specific Japanese influences, if any exist?

I'll give a barebones history of Korea to hopefully explain some of your question(s).

The idea of a distinct and independent Korea goes back to the 4th century BC with the ancient state of Gojoseon. After the fall of Gojoseon, the Korean peninsula developed its own Three Kingdoms period with Baekje, Silla, and Goguryeo with all three kingdoms claiming to be the legitimate successor to Gojoseon.

During the 7th century, Silla was able to conquer most of the Korean Peninsula with the help of the Tang. Most Koreans today are thought to be descended from the royal Kim family of the Silla kingdom.

Goryeo then succeeded Silla in the 10th century which became a vassal state under the Yuan in the 13th century. After the fall of the Yuan, Joseon succeeded Goryeo.

Joseon became a tributary state under the Qing dynasty in 17th century with the fall of the Ming. It was only until 1895 when China was forced to give up tributary relations with Korea by Japan.

The relationship between China and Korea is very complicated because of the role of China in Eastern civilization and the different states that made up "the Chinese".

Most Chinese dynasties maintained themselves essentially as the center of "civilization"; because of this, Chinese states refused to maintain any diplomatic relations other than tributary systems. (I put civilization in quotes because their definition of "civilization" is slightly different) They all spanned massive empires spreading "civilization".

The Korean peninsula was relatively able to maintain its independence from Chinese assimilation since Goguryeo's victory over the Sui. Except for the conquests of Korea by the Yuan and Qing, Korea remained relatively independent from China.

Also, I think it is unfair to consider Manchurians "fully-assimilated". Manchurians today are often not considered "Chinese" and have distinct cultures. Their "assimilation" or acceptance of Chinese culture comes from the fact that they ruled over China during the Qing dynasty. Many Chinese people saw them as foreign rulers and many Qing emperors adopted Chinese culture to appease their subjects.