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.

Godzilla in Fortnite is available as part of the Battle Pass and in the Item Shop, as well as a limited battle royale feature

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.