One thing that's interesting about scenes like this, where someone tries to reason with the Punisher, is how they reveal how much of his self-image has to do with being damned. He doesn't want anything, he doesn't believe anything, he's dead already as far as his feelings. It's just that he's still breathing and his body is still working. So while he remains, in his view his duty is taking as many criminals out of the world as he can. And even remaining alive is part of that duty. He doesn't have a death wish--that would imply some kind of desire. So he doesn't do it in order to get himself killed. He doesn't care if he is, but he has to do the best he can to keep himself alive anyway so that more criminals can be killed.
But it doesn't mean there's anything he wants, and there's not a way of looking at this that he's not already considered and is beyond, which is a way of thinking those who try to reason with him don't comprehend, and can't, being still psychologically "alive." Frank knows already, whatever it is you say to him--that his family's already avenged, that this is crazy, all that, he already knows, but it's not relevant to him.
In some ways it's the way Ditko's Mr. A can be made to work. Frank is as absolutely implacable in his integrity. Except with Frank, again, "integrity" seems too human, too pre-Vietnam, a word.