In "The Cask of Amontillado", why does Montresor feel as though he has the right to take matters into his own hands and kill Fortunado?
The crimes of Fortunato are no against society but social crimes against Montresor himself. Therefore, there is no other remedy available to Montresor other than forgiveness. Given the motto of his family, "no one injures me without impunity", forgiveness is not an option Montresor considers. The clue to this is in the first line of the story, "A thousand injuries of Fortunato I had borne as best I could, but when he ventured upon insult, I vowed revenge." In those lines, Montresor reveals that Fortunato has "injured him", probably by unkind remarks about his family and its place in society, many times. Montresor says he has remained quiet but Fortunato eventually insulted him and/or his family. Since Montresor is not one who takes an insult lightly, as witnessed by his family's coat of arms, he decides to take revenge.
Montresor is acting on an old Italian belief that revenge is an acceptable practice, even if the process of revenge includes the murder of the offending party. So Montresor's actions against Fortunato are perfectly acceptable in Italian society in this period.
Montresor is the offended party, Fortunato is the offender, the revenge is required and necessary so that Montresor's family name can be restored and his dead ancestors can once again rest in peace.
At the end of the story, he utters "rest in peace," meaning that once Fortunato has been punished his task is done. His relatives can return to their eternal slumber.