Can a king kill a king to get out of check?

No, if a white king for an example checks a black king, then that white king is in check because of the white king. That means that a king can never be checked by a king. Think of it as a forcefield, neither king can come within a 1 square range of the other.

The only possible way a king could check another is if it was blocking the path of one of its own pieces that would be checking the opponent's king, and the king moved to unblock the path. This is called a discovered check.