Montresor is a character created by Edgar Allan Poe to fulfill a purpose. He must be the kind of man who would commit a terrible murder. Poe has made him intelligent, proud, patient, cunning, deceitful, and cruel. His French name suggests that he is an outsider in the Italian city where he lives, which is presumably Venice. He lives in a palazzo, but he is all alone and has a hard time keeping up appearances and making ends meet. He appears to deal in expensive things like works of art, antiques, jewelry, and gourmet wines. He is an aristocrat but in constant danger of losing his social position if he fails to earn enough money.
It should be obvious that in many respects Montresor is like Poe himself. Poe may have written "The Cask of Amontillado" in order to express some of the hatred he felt towards some real person who had offened him. He had numerous enemies because of his petulant nature and his caustic literary criticism. Both Poe and his character Montresor seem like lonely, bitter, unhappy men, and both seem to have been overly fond of liquor.
As the narrator of the tale, Montresor is either confessing or bragging the murder that he committed years ago. This, right away, may be used to measure to what extent he may be a narcissist or a sociopath. He may be a narcissist because it is clear that his narrative is not an act of repentance, but a return to a scene in his life for which he does not take much responsibility, and of which he seems quite proud.
He could also be seen as a sociopath who cannot connect to others, is always alone, unhappy, and whose anger at his own state is transferred onto others, who are happier, or richer, or better suited to life than he is. This being said, narcissist and sociopathic are good descriptors for Montresor.
Stemming from those two descriptors, come the behaviors expected from those who have those mental conditions. They are all evident in Montresor. He is vengeful, envious, and so egotistical that he cannot let go of Fortunato's supposed insult.
A wrong is unredressed when retribution overtakes its redresser.
Waiting and pre-meditating Fortunato's death makes Montresor also a predator; a cold hard killer so reactive that he does not feel any worry about possible consequences. He has been lucky enough to get away with it. He knows that, too.
It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.
Stewing over what he feels is "insult", Montresor also comes off as obsessive, petty, and unreliable. Even though first person narrators are often unreliable, Montresor shows that he may very well be also a mad man, for which we cannot ascertain for sure whether his story is based on reality, or on what his mind dictated him at the time.