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.

On top of the Godzilla skin, you also have the opportunity to become Godzilla in Fortnite matches, check out our guide at that link to find out how and what cool things you can do when you transform into the monster.