To whom is the Duke speaking in "My Last Duchess"?
When the Duke refers to "The Count your master," this tells us that he is actually speaking to an emissary of a count: a representative that the Count has sent to broker the marriage of his daughter to the Duke, as the Duke also refers to the Count's "fair daughter's self," who he says is his "object."
However, it also seems likely that the Duke is sending a warning to the Count, and even perhaps the count's daughter. He has explained that he "choose[s] / Never to stoop." To him, it would be "stooping" to explain to his wife that she should value him and his very old family name above any and all other gifts she might receive. In his mind, she ought to know this, feel this, already. What frustrated him about his last duchess was her habit of being "Too easily impressed; she like whate'er / She looked on, and her looks went everywhere." She seemed as grateful for the sunset or some fruit as she was for the Duke's love, and he would not lower himself (what he calls stooping) by explaining to her why she should appreciate him the most. Rather than "stoop," he seems to have had her killed. He says, "I gave commands; / Then all smiles stopped together." Now, it appears that he is warning the Count and the Count's daughter, his potential future bride, via the Count's emissary, about how he expects to be treated by his wife and what he will and will not consider acceptable behavior. In other words, she will have to act as though the Duke is the best thing that's ever happened to her once they marry, or he might just get rid of her as he did the last duchess.
Part of the mastery of Browning's art in this poem is that we are only told who the audience of the Duke's dramatic monologue is at the end of the poem, after he has seemingly quite cheerfully narrated how he had his "last Duchess" disposed off because of how, in his perception, she bestowed attention of others. Thus, having established the immense pride and cruelty of the Duke, it is highly ironic that we discover in the last few lines that the silent listener is a representative from a Count whose Duke the daughter is negotiating to marry:
The Count your master's known munificience
Is ample warrant that no just pretense
Of mine for dowry will be disallowed;
Though his fair daughter's self, as I avowed
As starting, is my object.
Now, when we think about this, this is either incredibly ironic or/and it is incredibly chilling. Either we think that the Duke has no awareness of what the story of his last Duchess is doing to the listener, or, he is deliberately sending a message to the Count about the kind of behaviour he expects from a wife and the kind of response he can expect if his daughter does not behave accordingly.