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?

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.