The enotes to Gothic literature make mention of the element of "privileged irrationality and passion over rationality and reason." This chaos of irrationality seems to be what moves the plot of "The Black Cat" by Poe and is its most salient feature. Even the narrator suggests his own abnormality:
Hereafter, perhaps, some intellect may be found which will reduce my phantasm to the commonplace....Some calmer intellect, more logical, less excitable than mine will perceive...nothing more than an ordinary succession of very natural causes and effects.
That the narrator is "friends" with the cat in the beginning, yet he "maltreats" it by "a fiendish malevolence" in his "frame" which induces him to cut one of the cats eyes out points to his "privileged irrationality." In a chaos of confusion, the next day he feels some remorse for his act. Nevertheless, he subsequently hangs the cat
because I knew that it had loved me and had given me no reason of offense--hung it because I knew that in so doing I was committing a sin, a deadly sin that would so jeopardize my immortal soul....
After his house is ruined by fire, the disturbed narrator returns to the ruins to see the cat in relief: cast in what had been freshly spread plaster. This "phantasm" of the cat remains with the narrator, who takes in another black cat. One day, however, in the chaos of his mind, he becomes angered at the second cat who causes him to trip. With an axe, he attempts to kill the cat; instead, his hand is stopped by his wife.
Goaded by this interference into a rage more than demoniacal, I withdrew my arm from her grasp and buried the axe in her brain. She fell dead upon the spot without a groan.
This act is surely one of chaotic irrationality. It is horrific, too. When he cannot find the cat to rid himself of it, the narrator worries no more. Nor is he worried by the several visits by the police. Irrationally, the narrator boasts of his sturdily constructed walls, and a shriek is heard. Upon tearing down the wall on which the narrator has rapped, the police discover the decayed remains of the woman and the black cat whom the narrator describes,
with red extended mouth and solitary eye of fire, sat the hideous beast whose craft had seduced me into murder, and whose informing voice had consigned me to the hangman....
Gothic literature has several distinguishing features. It often focuses on the dark, evil side of human nature. It asserts that everyone has a dark side, and in these stories, we find the characters giving in to that dark side, and doing awful things. Also, gothic literature has elements of the supernatural or unexplainable to it; ghosts, paranormal activities, and malevolent hauntings. Gothic literature also focuses on the confusing complexities of the human mind; characters often get lost in their thoughts, and suffer breakdowns from mental anguish and instability.
If we look at the story, the narrator grew up a kind, loving and amiable man. He has a "docility and humanity" in his "disposition," and is kind to all people, and animals. However, the story takes a gothic twist when, because of "intemperence," or alcholism, his personality suffers a "radical alteration for the worse." He becomes violent, ill-tempered, and abuses both his wife and all of their pets. The narrator has given into his darker side, and it leads to the death of his wife, and his trying to cover it up. So, we see a narrator who embodies the gothic concept of "mankind's dark side."
Also in the story are elements of the supernatural. The narrator has disturbing visions of hanged cats, and his own cat inspired "terror and horror" within him. The black cat, supposedly disappeared, appears again, to torment the narrator and remind him of his evil deed, all while the cops are right there with him. So, the supernatural spiritual manifestation of the cat was occuring throughout the end of the story; this focus on ghosts and supernatural events is another gothic trait.
We see the narrator becoming lost in the maze of his own mind as the story continues, which is another gothic trait. He is tormented unusually and illogically by the black cat, and dwells on it so much that it drives him crazy.
Those are just a couple gothic elements to the story "The Black Cat" by Edgar Allan Poe. I hope that they help; good luck!