What do you think of the narrator? Is he madness or not?

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 19th 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 on 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.