"My Last Duchess" as a dramatic monologue.
At first, we do not realize how awful the Duke is. He mentions the painting of his late wife, compliments the painter of the portrait, and takes note of the fact that no one is allowed to move the curtain that covers her portrait besides himself. We might even assume, at this point, that the curtain exists because it is too painful for the Duke to look at the portrait all the time. His late wife's cause of death is a mystery.
However, once the Duke begins to describe the Duchess's "flaw"—that she was "too soon made glad" by any little thing, and he wanted her to think he and his "gift of a nine-hundred-years-old name" were the best things and not be as happy with a sunset, for example, as she was with him—we start to see the Duke as selfish and unappreciative of the beautiful spirit of gratitude and joy the Duchess had. Moreover, he displays his immense pride when he explains why he refused to tell her that he wanted her to appreciate him more than anything else, saying, "I choose / Never to stoop." He would have considered it beneath him to have to explain such a thing.
Finally, the Duke reveals that "[he] gave commands; / Then all [her] smiles stopped together." In other words, he had her murdered. We now fully understand what a monster he is: a man who killed his wife because too many things made her happy. Also startling is the fact that he doesn't consider the risks of telling the representative of his (potentially) future fiancée's family that he murdered his last wife for displeasing him; he sees nothing wrong with his actions. When the Duke moves on to discuss another fine work of art in his gallery, "Neptune [...] / Taming a sea-horse," we understand that, to him, his wife was just another possession, and because he couldn't fully possess her, he killed her and turned her life into a painting: something he could fully own and control (with his curtain). Further, he couldn't "tame" her when she was alive as Neptune does the seahorse in the sculpture, so he "tamed" her in death.
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.