In a blitz game, when my king is checked, I make an illegal move that checkmated the opponent and pressed the clock. Then my opponent captured my king and claimed victory. Capturing the king is not a legal move. Who wins?
This is a good question indeed and a somewhat bizarre situation, which is not fully covered by the Laws. As you mention the opponent has completed an illegal move and you claim that. The relevant parts of the Laws are articles
The easy case is if your opponent has made an illegal move (that has been claimed) before. In that case they lose the game.
If this is a first illegal move your opponent has made in this game, however, then the arbiter will give you one minute extra time, the opponent's last move is taken back and they have to play any other move that is legal. This is where things become complicated. Your opponent has been checkmated and thus there is no legal move they can play. According to article
5.1.1 they should lose the game but only if the move producing the checkmate was legal, which it was not. Therefore, since your opponent cannot make any legal move, the game cannot continue but it has not finished and there is no result. Quite bizarre indeed.
The following is a somewhat bizarre solution to a bizarre situation. If the opponent is satisfied with a draw they could ask the arbiter to watch the next move and then make a move that leaves both kings in check. Then the arbiter will declare the game drawn according to article
If the opponent does not want a draw then the only solution that satisfies the Laws is that both players agree to correct the illegal move that caused the checkmate. This is allowed by article
A.4.2. If they don't agree the Laws do not tell you what to do. In such situation, I, as an arbiter, would rule to correct that move nonetheless without penalizing the player who made it (i.e. you) since it was the opponent's responsibility to make a claim.
Current Laws of Chess are here.
It depends whether USCF or FIDE rules apply, and may depend on whether an illegal move was previously claimed.
Under the USCF blitz rules listed in my US Chess Federation's Official Rules of Chess 5th Edition, an illegal move results in a penalty of two minutes. There is a variation listed, however, where an illegal move results in forfeiting the game. In this case, capturing the opponent's king is listed as a way of claiming a win, rather than being its own illegal move.
The FIDE rules changed for 2018. It used to be that in blitz, an illegal move would allow the opponent to claim a win. However, this rule was changed, and now blitz is similar to normal chess in that a second illegal move by a player will result in a forfeit, and the first one merely results in a penalty.
As for which player can be penalized, it seems to be the player who took the king. See rule 1.4.1, which says "'capturing' the opponent’s king is not allowed." See also this article in which an arbiter says "As far as I can see, the player who completed the first illegal move is now entitled to claim the illegality." However, this article is from 2006 and isn't quite an identical situation; I can't guarantee that an arbiter today would rule the same way.