To be clear, you cannot die while playing as Godzilla--you'll always turn back into a normal player at the end of it, no matter what. From there, you'll play the rest of the match out as usual. You don't, by the way, get any rewards for having been Godzilla in a match, aside from the happy memories.

To get the Godzilla Evolved skin in Fortnite you need to earn 12 levels (essentially, level up by playing any mode). This skin is reactive, so once you get your first elimination in a match it will begin to glow and the glow will become brighter the more eliminations you collect for that round.