The duke in Browning's poem is based on Alfonso II d'Este, the fifth Duke of Ferrara who ruled from 1559–1597. His first wife was Lucrezia di Cosimo de' Medici. Although she came with a sizable dowry, the Medicis were considered under the d'Estes in social status. Alfonso abandoned her soon after their marriage, from some time in 1559 to the day of her death, April 21, 1561. While the cause of her death couldn't be clearly determined at the time, many suspected her to have been poisoned.
As the poem goes on, the duke seems to become more and more bitter in talking about his late wife, saying how she was "easily impressed" and seemed to flirt with any and every man she came across. He also comments that his "gift of a nine-hundred-years-old name" didn't seem to be enough for her. He is clearly saying here that she should have been grateful to have married into a name that was much more respected than her maiden name. Near the end of the poem, however, the duke seems to make one brief comment on her death: "This grew; I gave commands;/ Then all smiles stopped together". He doesn't elaborate or even hint at the nature of the "commands"; instead, he immediately changes the subject and asks the visitor to return downstairs with him.
Clearly, Browning is portraying the duke (at least his fictional representation) as the murderer of his last duchess. However, he sets it up in such a way that this is only briefly mentioned or hinted at in one line near the end of the poem. The piece is not focused on what the duke has done wrong, but rather on how the duchess gave her affections freely to so many other men, to the point of ingratitude and perhaps, infidelity. Because this darker notion is concealed so well in the poem, it not only fits the duke's mysterious and embittered character, but it also allows Browning to get away with saying what he truly believes happened to the duchess in an inexplicit manner. The duke in the poem is essentially admitting his crime, but he is very clever and cautious by only hinting at what happened, not giving everything away. Browning is stating his claim in a similar manner through how he has written the poem– he is hinting that the duke killed his wife, but the ending is subtle enough that it could stem multiple interpretations.