No, a King can’t kill a King in Chess. A game of chess can’t be played without a king. The king will always be there. Just take care that your king will not get checkmated.
When you lose a lot of material like pawns and chess pieces then your opponent gets an overwhelming material advantage and can get close to your king and will finally checkmate him. Once you are down in material you are going to lose slowly but surely.
So make sure that you do not blunder away pawns and pieces senselessly as this will make your opponent stronger and stronger and his chances improve to checkmate your king.
See above: A king controls the squares AROUND him. So you are not allowed to move your king right next to the opponent’s king. There will always be a square in between the kings which separates them. This means it is technically impossible for your king to kill the opponent’s king.
What can kill a King in Chess?
Nothing can kill a king. The king remains on the board always. You can give a check to the king with any piece or pawn, and if the king can’t move away from the check, or if you can’t capture the check-giving piece or pawn, or if you can’t move another piece or pawn in between, the king is checkmate.