This may be a stupid question, but I figure it's worth asking. I recently visited the Museum of History in Seoul and it really struck me that almost all artifacts dating a couple hundred years ago were in Chinese characters, not Hangul. At what point did they begin to adopt a national identity, or rather an ethnically and culturally separate idea from that of China?
The use of chinese characters in korea needs to be seen distinctly from the culture of the population because while chinese characters were adopted for official purposes (e.g. government records) and by the aristocracy, culturally koreans did not see themselves as chinese.
The point when socially it had the closest similarity was during the Joseon kingdom, primarily due to a deliberate transformation effort rather than any natural process. If you were seeing artifacts from within a few hundred years ago then it was most likely Joseon era artifacts that were on display.
When the Yi dynasty (1392-1910) took over from the Koryo kings and started the Joseon era it did so with assistance from the confucian literati. Partly for that assistance, and partly to sever the ties of the buddhist priesthood (at the time very corrupt) had with the government, the literati were permitted to undertake a reshaping of government and society along confucian lines. Prior to Joseon importation of some confucian government practices had been taking place since the three kingdom period but during Joseon the government was re-designed to follow strict confucian principles.
From a societal perspective an example that illustrates the drastic nature of what took place was change in the status of women. During the Koryo era and prior women had for the most part equal status as men. They had equal rights of inheritance, equal right to hold property, and the right to wed multiple husbands. At the time it was typical for men to move into the wife's household rather than the woman moving into the male household.
From the perspective of how people viewed themselves we can see a distinct patterns of behaviour going back to even the earliest koreanic polities that indicate that koreanic people's saw themselves as distinct from chinese peoples and viceversa.
Some of the earliest chinese intrusions into koreanic lands, the Han commanderies, were almost immediately put under pressure by the surrounding koreanic emerging states. Goguryo was particularly aggressive in its stance towards chinese incursions and from the beginning viewed itself as descendants of GoJoseon and therefore distinct from the chinese. Ultimately this pressure led to the commanderies being destroyed.
During the Goguryo-Sui/Tang wars the tactics of Goguryo, such as extensive scorched earth tactics to deny the enemy supply, extreme stubbornness in siege defences, and a refusal to surrender until broken, exemplify how the Sui/Tang incursions were seen as something beyond the normal warfare that occurred vs Baekje and Silla and something closer to a war for survival.
At the end of the three kingdoms period a Tang army attempted to take possession of former Baekje territory but the remnants of the Baekje and Goguryo armies joined with Silla forces to expel the Tang from the peninsula. This occurred despite Silla forces having fought against them with the Tang.
Spontaneous uprising of righteous armies to battle subsequent foreign enemies such as the mongols, japanese, manchu, and the japanese again shows a continuation of this ethnic self-awareness.
So to summarise, Joseon is most similar to what a sinicised society might look like but at no point did the people see themselves as chinese.
Sources
On the specific issue of the use of Hangul instead of Chinese characters, the latter were used phonetically to represent Korean words, with only some crossover into Chinese proper names and loan words. There is no linguistic similarity between Chinese and Korean.
Sejong, even though he was one of the most neo-Confucian rulers of the Joseon dynasty, interpreted his duty to the people to include developing a native Korean alphabet which fit its phonology and was easy for common people to learn. He was resisted by the scholar class, which is why you see Chinese in many documents up to the twentieth century, but that is the opposite of meaning that Chinese was widely spoken. The scholars used it to maintain a monopoly on written information.
(Go to the Hangul museum next!)