Kublai Khan is thankfully well documented in our history, to the point where some of his writings survive. Furthermore, the Yuan Dynasty kept multilingual records and with the aid of Tibetan Lamas developed the 'phags-pa phonetic script that was used as a standard for languages within.

As we all know, Kublai Khan was a Mongol and a high ranking one at that, meaning his first language was Mongolian. Mongolian at the time was undergoing a radical change from old to middle as we can note from comparing inscriptions and the language in the Mongolian Secret History. As such, he probably used mainly Mongolian for personal matters.

However, we have his poetry, which is written in the literary style of Chinese at the time. The Yuan dynasty and especially Kublai Khan legitimized themselves to the Han by appearing sinicized and adopting Han customs and language to an extent. Kublai Khan's poetry and court records indicate a strong grasp of the language. As anyone who studies China will know, however, "Chinese" is quite a nebulous termas there are dozens of mutually unintelligable languages within China. However, we can assume that the administrative language was the Beijing dialect that would evolve into Mandarin and form the basis for modern standard. It is highly likely his court was expected to know this and Mongolian as well.

The Yuan Khan's straddled the line between Mongolian, Tibetan, and Han culture with the most successful having fused and adopted elements from each. In order to properly administer China, they needed to master the prestige Beijing dialect, in order to conduct clan diplomacy, Mongolian was needed. Some, in order to emphasize they had the mandate of heaven, wrote in literary Chinese. It is possible that some knew classical Tibetan as well, as it flourished during this period.