Imagine a wealthy, powerful man who is in between wives. (There are probably men around just like this right now!) He is showing someone around his house, and he points to a portrait. He says, "That's my last wife." He might emphasize the word "last" as he says this. What would you think of this man? I would think he is someone who "collects" women as trophies or objects, someone who is now looking for a new object to collect. (In fact, today we refer to such wives as "trophy wives." There is a certain contempt about his attitude, pointing out that he is important enough to acquire women as he pleases. The fact that this is a duchess lends to our feeling that the narrator is wealthy and powerful. Without even reading the poem, the reader can glean a certain amount of information about this narrator.
My own father, capitalizing on this title, frequently introduced my mother as his first wife as a joke. The joke was that he could get rid of her any time he pleased and go looking for a second wife. (They were married for 62 years.)
The title of Browning's poem is very appropriate because he is referring to his murdered wife while shopping for a new wife in front of the duchess' portrait. The poem develops and the reader chillingly implies that the duke gave the duchess "commands;/ Then all smiles stopped together;" in other words, he killed his wife or ordered her to be killed by someone else.
The duke accuses his dead wife of flirting with any "officious fool" whom offered her "cherries" while she rode around on a white horse. This could be contrasting innocence (white) and sexuality (red). He further explains that she "thanked men," implying her sexual prowess with many. The Duke didn't believe that his wife was faithful, and he probably doesn't believe that beautiful women can be faithful.
In fact, the Duke believes that women are "object(s)" like his prized bronze artwork of "Neptune taming a sea-horse." The Duke is comparing women to both valuable art and a horse that needs to be broken. He hopes that his new wife will have a nice dowry like his first wife.
His objectified-female sentiments are obvious in that he adores the portrait more than the person the portrait represents. He sarcastically boasts "I call that piece a wonder, now" at the beginning of the poem. The piece is the portrait of the duchess, but also refers to her after death. She is appreciated as art post-humous, but not appreciated while living.