I don't think Matt believed he was God at any point. However, it was remarkable that Burton essentially had Matt pegged to a tee. I have always thought Matt was a bit selfish in the way he worships, which has obviously spilled over this season with his obsession with Kevin and wanting to be a part of this new religious experience. He has been fawning for Christ's acceptance his whole life despite the fact it has ruined his family. While he seems to be this great guy that is all about instilling hope in others through worship and prayer, i've always wondered if he really cared about creating that sense of hope or if it was more of a way to separate himself from the "common" folk in God's eyes.
When Burton was attacked by the lion, it essentially validated his belief that this Burton guy is a liar and God punished him for it, only exacerbating Matt's fanatical take.
Also, the actor playing Burton is the same actor who told Kevin to sing in last season's finale in the hotel. In essence, he resurrected Kevin and it looks like we are going to get another Kevin in the afterlife episode based on previews.