Checkmate with the King ?
You have to put a piece behind your king so by moving it you will create a discovered attack and if it is done correctly with coordination with another pièces you will achieve checkmate. That's how i got killer king.
You are talking about a Discovered Checkmate. This is where you move your King out of the way for a sufficient piece to deliver checkmate. Once you perform this action, you will get the Killer King achievement. In the following examples, the white King can move to another square to deliver checkmate to black.