Although this poem doesn’t actually have separate stanzas, we can analyze it by line number. Lines 1–13 set up the dramatic monologue, revealing that the Duke is showing an unidentified guest a painting of his late wife and responding to the guest’s apparent question about how such an “earnest glance,” a look of “depth and passion,” came to the Duchess’s face. In lines 13–21 the Duke darkly reveals that it was not his husbandly presence alone that brought to her cheeks that “spot of joy.” He complains that “She had / a heart.../ Too easily impressed,” and “her looks went everywhere.” It is interesting that the Duke had hired Fra Pandolf, a chaste monk, to paint his wife, hinting at a jealous nature. Lines 23–32 relay the Duke’s irritation that the Duchess was made happy by anything she saw as beautiful, and she gave approving looks and speech to anyone who was kind to her. The prideful Duke’s jealousy over her behavior is apparent in lines 30–34: “as if she ranked / my gift of a nine-hundred-years’-old name / with anybody’s gift.” From there the Duke explains to his guest that, although he could have made his disapproval known to her by saying, “‘Just this / Or that in you disgusts me; here you miss, / Or there exceed the mark,’” she would have used her wits or made excuse so as not to “be lessoned so.” Besides, he explains in lines 42–43, to tell her so would have been beneath him.

The big reveal comes in line 45, where he explains his solution to this problem: “I gave commands; / Then all smiles stopped together. There she stands / As if alive.” The guest is expected to understand exactly what the command was—the murder of his last duchess. Furthermore, in lines 49–53, we now understand just who the guest is and why the Duke has shared all this with him. He is an emissary of the Count, whom the Duke is hosting for dinner, where he will negotiate a marriage between himself and the Count’s daughter. So the story of his last duchess has all been a warning to be passed along. The Duke also suggests to the emissary in lines 49–51 that the Count’s generosity ensures that he will grant a large dowry along with the daughter, even if the Duke pretends not to require it. With further falseness, the Duke adds, “Though his fair daughter’s self, as I avowed / At starting, is my object.”

Imagine how creepy this whole conversation must feel to the emissary and how awkward it will be to pass along the veiled threat that the future duchess must follow the Duke’s wishes with exactitude or face certain murder. The Duke’s flippant boasting in lines 54–56 of another piece of artwork and its artist, as though the women being discussed are of less importance, clarifies that appearances are of the greatest value to this man. Women clearly mean nothing to the Duke beyond their ability to increase his status.

The first step in analyzing the poem is to look at its meter. Poetic forms are divided into stichic and stanzaic. Stichic forms are essentially continuous, rather than marked on the page or marked by rhyme as forming repeated multi-line units termed stanzas. The actual meter of My Last Duchess is heroic couplets, i.e. pairs of iambic pentameter lines that rhyme with each other. Characteristically of Browning, the lines often run over syntactically into subsequent lines, minimizing the natural pause on rhyme words, in  order to achieve a more natural and conversational tone.

That's my last duchess painted on the wall,
Looking as if she were alive. I call

The first couplet introduces the painting of the duchess as though she were a display object in a museum, rather than a recently deceased wife. This sets up the relationship of the narrative voice of the Duke to the Duchess.

The first person point of view and speech addressed to an anonymous audience indicate that the poem is a dramatic monologue.

That piece a wonder, now: Frà Pandolf's hands
Worked busily a day, and there she stands.

The second couplet introduces the act of creation of the painting. Again, we get a sense that something`s not right in the state of Ferrara, because the Duke seems more interested in the showing off the fame of the artist than in regrets for the death of his wife.