The conflict in "The Black Cat" is that of man against himself or, one might also say, man against alcohol. It is this internal conflict that creates all the external conflicts in the story and that propels the protagonist downwards into violence, murder, and perhaps insanity.
The way in which the two cats are employed in the story emphasizes this internal struggle. If the man had been in conflict with his wife all the way through the narrative, the reader might regard the conflict as being between the two of them. Yet the narrator's initial violence is aimed toward others due to the "Fiend Intemperance." Both of his ghastly acts against Pluto, the cat which he first disfigures by gouging out its eye, then hangs in the garden, take place during fits of drunken rage, and it is quite clear that the man's drunken self, not the cat, is the antagonist. The narrator becomes "wretched beyond the wretchedness of mere Humanity," and "evil thoughts became [his] sole intimates."
To further emphasize that the narrator's conflict is not primarily with the cat, he carries his mistreatment over to their second cat, which merely looks like Pluto. It is in trying to kill this cat that the narrator suddenly buries his weapon in the brain of his wife, killing her instantly. Although his malice against his wife, as against the cats, is fully intended at the moment of execution—making this murder—the conflict is, once again, clearly between his better and worse selves. Alcohol controls the worse self, allowing it to win easily whenever he is drunk and driving him to a "hatred of all things and of all mankind."