The poem My Last Duchess is a dramatic monologue.

The dramatis personae is the Duke and he could be Alfonso II of Ferrara. indeed, Under the title there is a reference to the setting of the poem: Ferrara.
So the poem is set in the duke's castle in Ferrara and the poet developed the monologue while the duke is in front of his dead duchess' portrait together with his messenger.

Right from the title you can notice the possessive adjective "My" that reveals the possessive personality of the duke about his wife. What more, the word "last" implies there was previous duchesses and it mean that there will be no more duchesses. Maybe, the word "last" could also refer to the last duchess' murder.

While the duke is in front the portrait, first of all he remembers the work of Fra Pandolf to paint the portrait, then he remembers the duchess' image through the element of the portrait.
After that he reveals his love for art but at the same time he declares duchess's unfaithful: she smiles to everybody in the same way.

Duchess died suspiciously so the declaration of her unfaithful could justify her murder and it gives to the poem an ambiguous and mysterious atmosphere.

The devices and the techniques used by the poet underlain it: the use of rhymes and enjambments reveals the chaos in the duke's mind and the flow of his thoughts.
Also the conversational speech reveals the confusion in the duke's mind caused by the murder: the direct speeches are directed to an inner listener who really is the mirror of the duke's personality.

So the poem is an example of Victorian dramatic monologue in which the poet expresses the dramatis personae's thoughts with critical distance.