Swing for him

Where did the phrase "swing for him" orginate.
Is it to kill him and then swing at the end of a rope or
swing an arm to punch him.

I don't know where it originated and, for a colloquial phrase like this one, you aren't ever likely to find out. The meaning is well-known though - at least here in the UK - and that's the second of your two suggestions.

"Take a swing at him" means try to hit him. "You'll swing for that" means you'll be put to death by hanging.