In "The Black Cat," why did the narrator kill the cat ?
Reading the story carefully reveals the answer to your question. After the narrator had become possessed by "the fury of a demon" and cut out the black cat's eye, the narrator explains how the "spirit of PERVERSENESS" overpowered him, causing him to do what he knew to be wrong. Note how he elaborates on this "spirit":
Yet I am not more sure that my soul lives, than I am that perverseness is one of the primitive impulses of the human heart--one of the indivisible primary faculties, or sentiments, which give direction to the character of Man. Who has not, a hundred times, found himself committting a vile or a silly action, for no other reason than because he knows he should not? Have we not a perpetual inclination, in the teeth of our best judgment, to violate that which is Law, merely because we understand it to be such?
Thus, the narrator argues that he killed the cat because of this spirit of perverseness, and how it encouraged him to do what he knew he should not, just because of man's natural inclination to do what we know goes against what is right. The narrator thus goes against his "best judgement" and chooses to "violate that which is Law" because of his perverseness, resulting in him finishing the job that he had started, and hanging his poor cat.