How does 'The Cask of Amontillado' relate to Edgar Allan Poe's life?

It doesn't. Many of Poe's contemporary and later critics ascribed to Poe many of the motives, thoughts and actions of the characters in his stories and poems. There is no doubt that writers write based on their personal experiences, but research has failed to connect incidents in Poe's life to actions in his stories. This story is a tale of revenge, murder and insanity. Poe did not exhibit thoughts of revenge, commit acts of murder or show signs of insanity. About the only connection between the story and Poe is the fact that the story title refers to a wine and Poe had his troubles with drinking.