This question covers a thousands year-long period of complicated history, which means I'll be summarizing broad stretches of history.

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Ancient, Medieval East, Early Modern Asia - China as the Principal Power

Understanding much of Asian history is understanding the "sinosphere" and influence China had on its neighbors and beyond. China from the start developed a sophisticated urban society and civilization, due to the geographical benefits of the fertile river regions of the Huanghe and Yangzi. The North China Plain is a relatively flat and fertile area which makes for denser populations. With the unification of warring states in the ancient period, a firmer Chinese identity formed from the Han Dynasty onwards. This Chinese civilization was understood by it's people as the center of the world (this is where the term 中国 (Zhongguo) "middle country" derives from), the bastion of civilization surrounded by "barbarian" societies.

A more concentrated population generally allows for a greater division and specialization of labor and skill, and thus enhance living conditions and develop technology. Of course, China was not exclusive within Asia to develop sophisticated culture, technology, society, etc, however China was certainly notable and influential as a great power in the region. They had a capable military, strong, unified, and large domains, access to resources, and a flourishing arts, crafts, and trade sector.

Korea is geographically closer to China, being situated on the mainland, than Japan, further eastwards as a series of islands. Early Chinese states such as Han or Northern Wei bordered the northern marches of the Korean peninsula, establishing commandaries that would lead towards greater and closer contact between the civilizations in the coming centuries. Maritime travel in this period was more limited, and sailing to and from Japan was not a regular occurrence.

The Tang Dynasty can be considered a major zenith period of Chinese history. It was here that more concrete diplomacy was conducted with Korea and Japan. The Tang Dynasty, allied with the Silla state, waged war against the powerful Goguryeo state, leading to its eventual downfall. In the aftermath of the conflict, territorial disputes led to costly wars between Tang and Silla.

In Japan, the relative stability of the Nara Court created a noble court culture that looked towards the advanced Chinese in the west. Poetry and literature flourished in the highest levels of Japanese society, and Chinese methods of bureaucracy, as well as the writing system, were also adopted. It was during this time that Buddhism landed in Japan from Korea, which would play an immensely significant role in society from then on. In the 9th century, a number of Japanese missions to the Tang court were conducted, bringing back models for dress, city layouts, class structure, philosophies, and more.

Corresponding to Japan's Nara court was the court system of the Silla. It too adopted many customs and models from Japan, and Silla's control of most of Korea also led to a stronger unification of native Korean culture and identity. Silla's decline led to Goryeo inheriting and establishing a unified Korea, which held off Liao and Jin incursions. Goryeo was a robust state that is remembered for its sponsorship of Buddhism, which had been introduced before from China during Silla's reign. Goryeo had extensive trade and communication with the Chinese states, in a form of semi-tribute. Goryeo's autonomy would last until the Mongol Invasion, and though it was not completely incorporated into the empire, it fell under heavy influence.

An indication of this was the Mongol military expeditions into Japan, with many of the navymen and soldiers being composed of Goryeo Koreans. Some negative sentiment against Koreans developed due to the understandably nationally shock of a military invasion. As an example, there was a certain folk saying which went 「蒙古高句麗、鬼来るぞ」"The Mongol-Korean Demons are coming!" to frighten misbehaving children.

The replacement of the Yuan Mongol Dynasty by the Ming roughly corresponds to the rise of Joseon in Korea. These two Dynasties had especially strong ties to each other, maintaining regular trade and diplomacy and both heavily espousing Confucian principals. The Japanese under the Ashikaga military dictatorship/shogunate sent a series of trade missions between the 15th and 16th centuries, until civil strife in Japan put an end to these envoy trips.

Toyotomi Hideyoshi's unification of Japan allowed for a brief moment of pause in the warring states period. However, the constant warring in the past several decades was a great burden to the new Shogunate government. It was feared the military society and unruly nobles might challenge Hideyoshi's new authority, so a national purpose was called for with an invasion of Ming China, the center of splendor and grandeur in the world. Geographically Korea was required as a stepping stone onto the Chinese heartland, and the Joseon naturally refused an entire military to pass through its realms, leading to the two major conflicts between the Korean and Japanese in the late 16th century, the Imjin War. There are records of Japanese atrocities committed during these invasions, the Japanese way of war on their homeland at times involved complete annihilation of enemy clans. However, it should be stated many of the Ming reinforcements were just as brutal against the Korean common classes. Regardless, many Korean artifacts and artisans were taken and relocated to Japan in this time. These invasions are national scars in Korean history, which correspond to future events that leads to Korean nationalism.

Following Hideyoshi's death, the costly invasions on Korea were called off and internal security issues needed to be addressed by the Japanese. Tokugawa Ieyasu's adept maneuvering of political alliances allowed him to topple the Toyotomi establishment and build a new Shogunate. Henceforth we see a shift in China, Korea, and Japan towards isolationism. In the previous century, the Ming trade expeditions were already canceled due to their cost and a need to address domestic matters. Korea and Japan limited contact in the west due to the threat of Christianity. Aggressive Dutch attempts at opening trade against the Ming were successfully repelled. Poor administration, mismanaged military, and economic failure enabled the Qing to assume control of China in the 17th century. The Qing Dynasty, although militarily successful in conquering large swaths of territories westwards, had to deal with the management of such immense domains. Focus simply favored internal management over foreign policy.

Modern History - Rise of Japan and the Impact of Imperialism

In the mid 19th century, American and European powers forcibly opened up Korean and Japanese ports. The arrival of American Matthew C. Perry's ships onto Japan shook the nation. Over the course of the following decade, various Japanese lords and domains adopted Western elements as the nation opened up. Anti-foreign sentiments grew, and the Shogunate was deemed ineffective with addressing the western powers and containing regional conflicts. The Shogunate lost power in 1868, and power was "restored" to the Imperial Family. Japan thus began a period of rapid modernization, modeling much of its nation-building on Western nations. France and Germany were modeled for the army, with England for the navy. Japan understood it needed to be strong in order to fend off Western domination of Asia. The Opium Wars in China were alarm bells which Japan wished not to suffer itself. As Japan's plans for Imperialism grew and widened, as we shall discuss in the next paragraphs, Japan's rationale changed and evolved too. As Western imperial powers moved into Asia, Japan not only wished to maintain its absolute autonomy, it justified manage of its aggressive diplomatic and military invasions as a way to "save" fellow Asian nations from the western sphere of influence. China was deemed to have degraded, and it was the responsibility of the Japanese to set Asia on the right path, even if force is necessary to achieve those means. Japan as a nation wished to be respected within the world, as a fair player and equal power to America and the West. By matching the western powers, Japan needed to modernize and westernize, adapt the best qualities of the west, and make a stake for themselves so no other power could challenge them.

In 1869 the new Japanese government sent envoys to Korea to establish diplomatic ties. A sort of miscommunication occurred, as the Japanese used the characters for "Emperor" in their message, referring to the restored Meiji, which offended the Koreans who at the time closely-held ties with Qing China and recognized the Qing Emperor as the emperor. Debate occurred in the Japanese court about punishing Korea, which eventually was rejected. Saigo Takamori, of the Satsuma Domain, a leading power in restoring the Imperial cause, had hoped to use an invasion of Korea as a sort of honorable sacrificial mission for the samurai class to meet a noble end in the face of the vanishing power during Meiji's reforms. Although Satsuma would later rise up in opposition to the Imperial authority, their rebellion was crushed. Regardless, Japan began to make designs for expansion. Its own conflicts were costly and industrialization and modernization required funding and resources, the latter of which Japan as an island nation lacked.