The most likely explanation is that this story is a translation of an old letter written in Italian or French to a friend and found among some papers after that friend's death. Or it could have been a letter that was never sent and was found among Montresor's papers after his death. The event described could have taken place more than fifty years ago, because the letter itself could be very old. Montresor himself must have been quite old when he wrote this confidential letter (assuming my theory is correct). If he were actually speaking to someone, such as a priest or an old friend, the story would lose some of its verisimilitude, because the story is in English and Montresor is a Frenchman living in Italy. The story reads more like a document than like a spoken narrative. It is similar to Poe's "Manuscript Found in a Bottle" in its fictional format.
There's been much speculation on whom Montresor may be addressing, fifty years after the fact. The second sentence reads, "You, who so well know the nature of my soul, will not suppose, however, that I gave utterance to a threat," which seems to imply Montresor is addressing his priest, a person who would well "know his soul," and to whom Montresor would not want to appear like a threatening person, especially since it is likely that--given the time span--he is now confessing his sins on his deathbed. This reading also lends a nice irony to the last sentence of the story,"In pace requiescat!" (Latin for "In peace may he rest.")
he means you as the reader
I am not sure of all the answers, but what I do know is that Montresor is telling the story 50 years after it happened. He also seems to know the person he is telling very well, as he says "you" knows the "nature of my soul."