In "The Black Cat," what do you think is the relationship between the narrator and the black cat?

We are given no clear answer to this excellent question.
Rather, the text seems to tantalise us by pointing towards a supernatural explanation of
the relationship between Pluto and the narrator, and what transpires with the black
cat's apparent reappearance after his murder and how the black cat makes sure that the
narrator pays the price for his crime. Note what the narrator tells us about the black
cat towards the beginning of the story:

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In speaking of his intelligence, my wife, who at
heart was not a little tinctured with superstition, made frequent allusion to the
ancient popular notion which regarded all black cats as witches in disguise. Not that
she was ever serious upon this point--and I mention the matter at all for no better
reason than that it happens, just now, to be
remembered.

The casual way
with which this fact is narrated perhaps underlies the subtlety of the supernatural
suggestion that it gives rise to. The way in which the black cat reappears, even after
being murdered, and the other apparently supernatural events that accompany it, would
indicate that the narrator's wife was right in her suggestion that is so quickly
dismissed by the narrator. The black cat was a witch in disguise that determined to make
the narrator pay for his act of murder.