Yeah, I think it would be better to first explain the liar's paradox to give the broad brush strokes, and then go into the details of Gôdel numbering.

It seems like most expositions of Gödel's incompleteness theorem go into a surprising amount of detail about Gödel numbering. In a way it's nice though, because you see that the proof is actually pretty elementary and doesn't require fancy math as a prerequisite.