The narrator of "The Black Cat" is what's called an unreliable narrator. This means that we can't be sure that what he's saying is true; we simply can't take him at face value. Instead, we need to read his words carefully and see if we can construct a plausible account of what happened. Very few of us are trained psychologists, so we can only surmise as to his true mental state.
The narrator starts by telling us that he isn't mad; but he does acknowledge that some degree of psychological disintegration has taken place. We sense that the narrator is reluctant to acknowledge madness because of the shame and stigma that such an admission will bring. In reading the story, we need to remember that people with mental health issues in the nineteenth century were often treated appallingly by society, so we can understand why the narrator doesn't want to be tarred with the brush of insanity.
Yet the narrator undermines his own case by his meticulous account of the killing of his wife and the black cat and the events leading up to their gruesome deaths. He says he isn't mad, but then convinces us otherwise by the horrific story he relates. What is relevant to this particular question is the fact that both killings are carried out in the spur of the moment; there's nothing premeditated about them. This would seem to indicate a sudden lapse into insanity on the part of the narrator.
There are other signs of madness in the story. The narrator has a remarkable ability to stand back and watch his own mind disintegrating from a distance. When he begins to hate the second cat he took home with him, he's aware that he's starting to lose his feelings of humanity, about which he made such a big deal earlier.
The available evidence, therefore, would seem to suggest the presence of at least two key components of insanity in the case of the narrator. Madness isn't simply a condition that's distilled into a sudden, impulsive act; it's also a process that effectively separates the insane individual from their own mind. That's what we mean when we say that someone has "lost their mind." And that's what appears to have happened here.
The opening lines of "The Black Cat" by Edgar Allan Poe are spoken by the narrator of the story, and he insists he is not insane; however, there is sufficient evidence to make the case that the narrator is suffering from some kind of mental illness.
He writes the story from jail, and this is how he begins:
For the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence. Yet, mad am I not -- and very surely do I not dream. But to-morrow I die, and to-day I would unburthen my soul.
This is the night before the narrator is going to lose his life as punishment for the crime of killing his wife, and he assures us he is only telling the "homely" story because he wants to unburden his soul. There are two motives for such an unburdening, and it determines whether his account is truthful. A condemned and guilty man (which is what the narrator is) may wish to write a confession of his actions, without any justification or attempt to mitigate his actions; or such a man may wish to write his version of the story in an attempt to garner sympathy from his audience and excuse his own behavior.
This narrator tries to excuse his actions by claiming that "from my infancy I was noted for the docility and humanity of my disposition." This obvious exaggeration is clearly an attempt to make himself the victim in his wife's murder. He tries to blame his actions on a cat--two cats, in fact--and on his wife; yet he demonstrates a blatant tendency to intemperance (impatience) and overreaction after several years of contentment with his wife and Pluto.
The truth is that he underwent some kind of change which precipitated Pluto's subsequent change in temperament. The narrator's change came first:
I grew, day by day, more moody, more irritable, more regardless of the feelings of others. I suffered myself to use intemperate language to my wife. At length, I even offered her personal violence. My pets, of course, were made to feel the change in my disposition. I not only neglected, but ill-used them.
His mental state is questionable here, and the symptoms are indicative of some kind of mental illness. From this point, the narrator's actions are all affected by that illness. Is he insane? Perhaps, but he certainly suffers from some kind of mental illness.