Let's begin by thinking about the ways in which the psychological approach examines literature. Literary critics using psychology explore the motives, behaviors, and psyche of the characters as well as the psyche of the author to discover deep meanings in a work of literature.
To apply a psychological perspective to Edgar Allan Poe's "The Cask of Amontillado," critics might begin with the author himself. Poe lived a dark, mysterious, and lonely life filled with grief, failure, and substance abuse. His troubled psyche certainly appears in this story, at least in the darkness of the tale, if not, at least a little, in the single-minded madness of the protagonist. Psychological critics would try to relate Poe to his character and see how much of the character's madness comes out of the author's own disturbances, desires for revenge, and dark thoughts.
Further, psychologically based critics would be fascinated by the madness of Montresor. They would strive to figure out the motives and disorders that would lead to a man walling up another man in a dungeon and leaving him there to die a slow, miserable death. Indeed, Montresor is the perfect character for a psychological analysis that would closely examine his background, the nature of the insults that sent him over the edge, his paranoia, his meticulous planning, and his murderous intents.
The psychological approach would also pay close attention to Fortunato, who goes crazy in quite a different way. Critics would examine first Fortunato's arrogance and pride as well as his treatment of Montresor and the reasons behind it. Then they would look closely at his descent into madness as he descends into the catacombs. Finally, they would describe and interpret the effects of the final stress on Fortunato's mind as he is walled up and left for dead.
Overall, psychological critics analyzing "The Cask of Amontillado" would focus on what drove Poe to write the story and what drives the characters to think and behave the way they do.