The most fruitful avenues for a research proposal on "The Cask of Amontillado" are probably enquiries into the historical background of the story. Here are two possibilities. It is often claimed that Poe's inspiration for the story was the grisly death of Lieutenant Gustavus Drane, who was sealed up alive in a vault by some of his fellow officers after he killed one of his more popular comrades. Poe heard this story while he was in the army at Fort Independence in 1827. You could explore the events surrounding Drane's death and how far the details correspond with those in Poe's story. You could also look at alternative literary sources, including the work of Thomas Dunn English, a writer with whom Poe had a long and bitter feud. Both used literary parody to attack the other, and the allusions to the masons and to Montresor's coat of arms in "The Cask of Amontillado," for instance, appear to refer directly to English's work.
Another method of approach, would be to look at the tradition of macabre revenge stories set in Venice. Although the location of "The Cask of Amontillado" is never mentioned, all the incidental details, from the drunken carnival to the damp catacombs, suggest that it is set in Venice. This would certainly be apposite, since the Venetians had a singular reputation for pride and cruelty in the nineteenth century. Reuben Parsons wrote in his popular historical work, Some Lies and Errors of History (1893)
To the average mind the history of Venice is a bloody and lurid melodrama. Dungeons under the canals, cells exposed to the fury of an almost torrid sun, hidden doors ever menacing an egress of spies and assassins, virtue and valor ever succumbing to dagger or to poison...
You might research the medieval and renaissance stories on which this reputation was based, and how far Poe drew on them, either for events or atmosphere.