When in the Bible it talks about John the Baptist being Elijah, it is talking about Elijah's spirit being upon him, the same gifts, he was the new Elijah.
And there is a verse, but I don't remember where it is, (I forget where verses are a lot) but I'll ask my pastor when he gets back from a camp what verse it is, but this is how it goes:
"For it was commanded to man to die once and then be judged."
That says we die once, and then we are judged. So we can't have multiple lives, and as soon as we die, we are judged to Heaven or Hell (even if we go to Heaven we are still judged for our deeds).
So there you go, no reincarnation.