At the center or heart of the dramatic monologue, "My Last Duchess," the Duke is a snob and the Duchess was not. That's why he has her executed.
The Duchess is now in an ideal state, according to the Duke's thinking. She is now (in the poem's present) a work of art, pretty and idle, doing nothing but reflecting back on his good taste and good name. The work begins and ends with works of art: the Duchess' portrait and the sculpture of Neptune. That's what the Duke wants for a wife--the ultimate trophy wife: stand there and look pretty and adore the Duke.
There's no suggestion of a lover or that the wife did anything wrong, according to normal standards. She simply was pleased with people and animals and elements of nature. In short, she was pleased with and courteous to that which was not the Duke. And the Duke couldn't stand this. His wife must reflect back on him like a work or art, a prized possession, a rarity. Anything else is insubordination and ingratitude.
And, of course, the entire monologue is an implied threat--if his new wife doesn't behave as he wishes, the same thing will happen to her.
Talk about opposites!
The Duke is the epitome of a jealous, obsessively possessive husband. He is rich and powerful and comes from a very aristocratic family. He is proud of himself, his wealth, his achievements, and especially of his family "name" that he so graciously "bestowed" upon his wife, who did not appreciate that he allowed her into his esteemed family. He is conceited. He tells his visitor of the many "imperfections" his wife had, as if he was entitled to more -- to a perfect wife. By discussing his wife's portrait in such a callous way - "that's my LAST Duchess" there on the wall, he will soon have another wife and perhaps will have many others if they don't measure up. His "LAST" duchess was merely a possession as evidenced by the fact that he keeps her portrait hanging there. Too bad she didn't act more like a possession. Too bad she esteemed others more than he. Did he talk to her about it, though, so she could make some changes? Of course not! Why should he stoop to this? So, he was no doubt non-communicative.
We have to read between the lines to figure out what the Duchess was like, but Browning skillfully allows us to do this through the Duke's dramatic monologue. He reveals what a cad he is and what a much nicer woman his wife must have been. The "Last Duchess" was apparently flighty in the opinion of the Duke. The blush on her face, he says, was because she was thinking of a lover, or perhaps the artist painting her was her lover. No matter, he was not he that caused that blush on her face. Although the Duke sees these as faults, the Duchess was "innocent, too easily pleased and impressed." She could enjoy the sunset. He feels that she should have enjoyed being his wife more than looking at the sunset. She apparently was very polite and respectful to all people, but the Duke interprets this as her lowering herself.
Naturally, the Duke could not allow the Duchess to live. He gave some orders, and all of her smiles stopped. And now, he is shortly to meet with someone else to secure a new wife.