How would you use this word in the "he's the best" sense? Αυτός/ή/ό είναι κορυφή;
There is a sentence like that in the course, but it's more common to say "αυτός είναι κορυφαίος" or αυτός είναι στην κορυφή".
So can this also mean the top of something? Like Ο παππούς μού βλέπει την κορυφή για το βουνό?
Yes, that would be "η κορυφή του βουνού", though. It needs the genitive case, as it is the top of something, as you said.