A critical analysis of work of literature examines literary devices, both (1) literary elements and (2) literary techniques. An analysis of any piece of literature contains information on these points, unless your assignment gives other specifics. For this answer, I'll assume a standard critical analysis; you can adapt the answer to fit the specifics of your assignment otherwise.
In critical analysis, the literary elements of a narrative are those that are predetermined, that must be present in narrative. These include characters as the agents of action; plot as the vehicle of conflict; structure, including chronological development, as the framework through which the narrative is seen; theme as the purpose of the narrative.
For an analysis of Poe's "The Cask of Amontillado," some of these points would be analyzed thus. (1) There are two characters, two agents of action: the protagonist, Montressor, and the antagonist, Fortunato. The characters are interesting because the protagonist has claim to this categorization because he is the main character featured and the driver of all the action. He does not have claim because he is in any way admirable or noble or good: he walls Fortunato up with a wooden barrel--a cask--of Amontillado wine to face his death alone behind brick.
(2) Similarly, the hapless Fortunato is the antagonist only because of the grievous wrongs, "[t]he thousand injuries [and] ventured ... insult," he has committed against Montressor. Thus his role as antagonist centers solely on Montressor's perception. (3) The chronological structure of the story puts it in a "frame" in which Montressor tells the tale in retrospect by narrating from some time in the future about the event in the past.
In critical analysis, the literary techniques of a narrative are those that are optional choices made at the author's discretion. Some of these are metaphor, personification, rhyme, irony, voice, point of view, and symbolism.
For an analysis of the story, some of these points would be applied thus. There is a good deal of ironic symbolism in the title and in Fortunato's name. The "Cask" of the title symbolically stands in for Fortunato's death cask, word play representing casket. His final circumstances (being walled up in the catacombs of his enemy with a cask of wine) ironically belies both the symbolism of his name (Fortunato: fortune, good luck, prosperity) and his successful life up to that point: "he was a man to be respected and even feared ...."
The first-person point of view allows Poe a voice that is immediate in proximity and replete with Montressor's thirst for revenge and his doubtful sanity. For further critical analysis, follow a similar pattern of reasoning while you consider the other points in elements and techniques that constitute Poe's composition of "The Cask of Amontillado": mood, tone, metaphor, figures of speech, theme, etc.
This short story, considered to by one of Poe's most famous, caused a bit of a stir when it was first published in the 1850's.
Consider some of the themes present in the story, how the story is structured, and the relationship between the two characters. Who are these two men? Why are they together? What was their prior history? The story structure tells you something about the character narrating the story. Why is he telling the story? Themes can include the concepts of retribution and/or revenge; for what reason(s) is the one character seeking out these upon the other? The link should provide some additional insight to help you get started.