Sometimes I just feel like throwing a bit of anarchy into whatever game I am playing. What are the rules for one player character attacking another player character? How does the battle play out?
There are no rules for players attacking other players, not to say you can't devise your own to make the game even harder to win cooperatively than it already is. I would imagine you might want to draw those lines at the outset, have it discovered in secret, or have it revealed to all by chance mid-game. Arkham Horror is pretty complex in itself so you'd want to think about several cases like how your character would assist the dark side, but it would be interesting narratively.
Have fun with it, but the answer to your question remains that there are no official rules for attacking one another. (It would be fun to play the antagonist and move a few monsters or choose between 2 mythos cards to inflict upon them, though, to spice up the flow of the narrative.)
In fact there is a possibility to play against other players.
In Dunwich Horror expansion there is a mission card Joining the winning team. You have to sacrifice allies in some locations. If you manage to do this - you win the game, and others loose.
If you like to introduce some variant in your game, you can try the Arkham Horor League.