In this poem, the only clue we have to guide us as to what happened to the first duchess is one little tiny phrase. And even that isn't super specific; we are left to infer, or fill in the holes by guessing. If you look closely at the text of the poem, there is a representative of a second wife who has come to discuss the terms with the Duke, and the Duke shows the representative the picture of his first wife. He tells the guy that she was super easily pleased, smiling and laughing at everything, which made him mad, because she was supposed to only smile for him, and show him the respect that his "gift of a nine-hundred-years-old name". After mentioning his anger at her supposed disrespect of him, he throws in this line, which is our clue: "I gave commands; Then all smiles stopped together." So, he "gave commands" and she stopped smiling forever. This hints that he told someone to kill her. He commanded someone under his power to murder her; that way she could never smile at anyone ever again. Her portait only can smile, and he has control over who sees it. It's a rather twisted and evil thing to do, especially because he is telling the new guy about it. He is basically confessing that he killed his first wife, as a kind of threat to the new one--"you'd better respect me, or you'll end up like her."
I hope that those thoughts help; good luck!
"My Last Duchess" a subtly patterned poem in pentameter that steps into the next line is the dramatic monologue of the Duke Ferrara as he negotiates a new marriage with the emissary for another wealthy family. As the Duke passes the portrait of the young Duchess who has died, he mentions her with less than regret to his guest that the painter Fra Pandolf made "by design when he portrayed "that picured countenance." Continuing his narrative, the Duke tells the emissary that Fra Pandolf
chanced to say, 'Her mantle laps/Over my lady's wrist too much,'or, 'Paint/Must never hope to reproduce the faint /Half-flush that dies along her throat.'
As the Duke's monlogue about the painting continues, it becomes apparent that the young woman's "looks went everywhere." When the Duchess
thanked men--good! but thanked/Somehow--I know not how--as if she ranked/My gifft of a nine-hundred-years-old name/With anybody's gift
the Duke is too insulted to excuse her and chooses "Never to stoop. He gives his wife "commands," but she ignores them. So, "all smiles stopped together." And, in the same breath, the duke nonchalantly says, "There she stands/As if alive" and continues his business of a new marriage without missing a beat of the pentameter. The Duke dismissed her life just as he has dismissed the painting. And, since this poem's setting is the Renaissance, the assumption by the reader must be that the Duchess has been killed since divorce in Renaissance Italy was nonexistent.