He was aware that he was setting up the Yuan as a Chinese* dynasty. To quote from Defending Heaven: China's Mongol Wars, 1209-1370:

Qubilai would later take a dynastic name for his reign and the Great Yuan would be brought into existence in 1271 as the dynasty that should rule all of China… he also made the claim to tbe the universal ruler of China, though without a dynastic title, in 1262… Qubilai was almost entirely dependent on China for all the war resources he required for victory over the Song. He could never again hope to rely on the Mongolian steppes to supply his army's needs, as the Mongol world empire was now largely a fiction.

For Qubilai (Kublai), ruling China was a much more reasonable long term plan than assimilation into a larger Mongol empire which, at that stage, no longer really existed with any real cohesion.

Someone with more appropriate flair and background in the Mongol Empire can come and offer a much better answer, but at least for now this should get you started.

^*that ^is, ^a ^dynasty ^in ^the ^Chinese ^tradition, ^not ^an ^ethnic ^Han ^dynasty

As Keyilan said, Kublai Khan intended to form a Chinese dynasty. Under his authority the dynasty name 'Yuan' was chosen (by a Chinese scholar), which is Chinese for 'Origin'. He appointed Chinese advisors to advice him on Chinese culture and court rituals. He also largely adopted the Chinese administration.

However, Kublai did not forget his title of Khan or his Mongolian roots, and neither did his successors. The Yuan dynasty always tried to keep a balance between their Mongolian and Chinese subjects. They basically had a dual identity. To their Chinese subjects, they were the Yuan emperors. To their Mongolian subjects, they were the Great Khans, heirs of Genghis Khan.

The Yuan capitals are a good example of this dual identity. Kublai made Beijing the capital of China, and build his palaces according to Chinese traditions. The city itself was build like any other Chinese capital. However, the location of Beijing shows his Mongol roots. Beijing is located very closely to the Great Wall, the border between 'civilized' China and 'barbarian' Mongolia. So Beijing is quite literally on the border between Chinese and Mongolian civilization. Other Chinese dynasties had located their capital more to the south, closer to the centre of Chinese civilization.

Besides Beijing, the Yuan dynasty had a second capital, Shengdu. Shengdu was located on the other side of the wall, though still close to Beijing. The Yuan spend the summer in Shengdu and the winter in Beijing. In Shengdu they hunted and followed Mongolian traditions. The emperor often spend the night in a tent, and his bedroom was also decorated like a tent. In Beijing they largely followed Chinese rituals and acted as Chinese emperors, although they still often showed their Mongolian heritage.

So, to answer your question: yes, Kublai Khan knew very well he was founding a Chinese dynasty. But he wasn't a 'Chinese emperor OR a Mongol khan' but a Chinese emperor AND a Mongol khan.

Sources:

George Qingzhi Zhao, Marriage as political strategy and cultural expression: Mongolian royal marriages from world empire to Yuan dynasty

Herbert Franke, China under Mongol rule

Sorry that I'm late to this, but as usual /u/keyilan and /u/lukeweiss give fantastic answers. I'd still like to add a little bit in terms of how the government was actually structured during the Yuan though, as it's a bit of a departure from previous iterations, but still retains the same frame.

The population of China itself was divided into four major groups: the Mongols, the "Central Asian Auxiliaries" (Naimans, Uighurs, Turkish peoples, and Tanguts), North Chinese, and South Chinese. The Mongols themselves remained the ruling class, while the Auxiliaries took over the roles of government workers, soldiers, and merchants. Additionally, since the Mongolians were often not literate in Chinese, much translation work was taken up by the Uighurs in bilingual government offices. So, while the government structure and reforms were modeled after Chinese dynasties, there was very little Chinese involvement in the government (it was more of a martial rule, as Kublai Khan established the emperor as more of a monarch). Another problem is that the Southern Chinese especially were strictly forbidden from engaging with many aspects of society under Mongolian rule. They were forbidden to marry Mongolians, carry arms, and at certain points forbidden from learning Mongolian or any other foreign language.

A passage written around the late Yuan/Early Ming attributes the fall of the Yuan to the distancing of literati and corruption of central asian clerks (bracketed text added for clarification; I've also cleaned it up a bit, as my reference uses Wade-Giles for phonetics, which I am not a fan of):

That Kublai was able to achieve the great unification of the empire was because he used authentic scholars [Ethnic Han literati]. He used authentic scholars to gain the empire, but he did not use authentic scholars to govern the empire. After more than eighty years, suddenly disaster has broken out. This is entirely because petty clerks are responsible for affairs. From the capital to distant areas, in the highest offices ... and in the lesser offices ... as well as in all the various minor offices, there are none that are not like this. Even if one scholar were to govern, how could he rectify these abuses? ... Therefore, I say that that which has spoiled the empire and the dynasty consists of the crimes of the clerks.

This quote and more info can be found in Mongolian Rule in China: Local Administration in the Yuan Dynasty by Elizabeth Endicott-West