You don't ask a question in your question, so I assume you just want the details of "My Last Duchess" as a dramatic monologue.
The poem features one speaker speaking to a silent listener in a dramatic situation. Though the listener is identifiable (fiance's father's representative), he doesn't speak. The purpose of the poem is characterization of the speaker.
In Browning's poem, the Duke is speaking to the agent, and reveals himself to be pompous, arrogant, warped, and murderous. The dramatic occasion is the negotiation of the amount of the dowry of the fiance, if you can call it a negotiation, since the Duke does all the talking/threatening.
Dramatic monologues are known for their implied characterization. In this case, the Duke doesn't tell the reader what an ass he is. The reader figures that out on his/her own. The threats made by the Duke are also implied, though they are very real.