Chaitanya left eight verses. That is the whole of his written output, a short work called the Shikshashtaka, and it says nothing systematic: it is about the effect of the name, the wish to be humbler than a blade of grass, and the pain of separation from Krishna.
The theology of the tradition was therefore written by other people at his direction. He sent six of his most learned followers to Vrindavan to recover the sites associated with Krishna and to compose the doctrinal and ritual literature, and those six Goswamis produced the body of Sanskrit work on which Gaudiya Vaishnavism rests. Its central formulation, inconceivable oneness and difference, holds that the soul and God are simultaneously identical and distinct in a way reason cannot resolve, which preserves both the non-dualist insight and the devotional relationship that requires two parties.
His followers also hold something about him personally: that he is Krishna, having taken the position and the emotional state of Radha in order to taste the love directed at himself from the other side. That claim is internal to the tradition and the atlas records it as the tradition's own.
The lineage continued through the Gaudiya Math in the early twentieth century, and from that root came the movement founded in New York in 1966 that carried the public singing of the name into cities across the world, which is how most people outside Bengal have encountered anything of him at all.