In Browning's "My Last Duchess," clues to the nature of relationships with regard to the Duke (the speaker in the poem) can be found in the poem's many verses—and its title.
The title, "My Last Duchess," indicates that this is the Duke's former wife. My "last" could infer that he has had more than one wife. However, the "last" is the subject of the lifelike painting that he is showing off to his visitor, identified as "the emissary." (His function will become clearer at the poem's close.)
While the painting is lovely, the Duke does not express any words of love or longing for this seemingly remarkable woman. Instead, he speaks first of a perceived curiosity of those who see the painting (including his guest) as to what inspired the passionate look upon her face—that the painter caught so precisely—but none would dare ask:
Strangers like you that pictured countenance,
The depth and passion of its earnest glance,
But to myself they turned (since none puts by
The curtain I have drawn for you, but I)
And seemed as they would ask me, if they durst,
How such a glance came there...
The Duke is possessive, noting that only he can draw the curtain from her portrait to show others: he alone has that control. Then he says:
Sir, 't was not
Her husband's presence only, called that spot
Of joy into the Duchess' cheek...
The Duke infers that the Duchess may have been unfaithful—once again showing his jealousy. There seems no proof that such was the case, but he is insecure (or smart) enough to believe that she did not look to him to get that "deep, passionate and earnest" look on her face.
The Duke insults his wife again by saying that even compliments paid to her by a [lowly] painter would make her blush—his insult is that she was too easily impressed by the words of others, even words that he insists are nothing but "courtesy" on the speaker's part. And then...
A heart -- how shall I say? -- too soon made glad,
Too easily impressed; she liked whate'er
She looked on, and her looks went everywhere.
While this is complimentary, the Duke uses it to vilify his wife in saying that she was too easily pleased. However, rather than detracting from her character, the Duke shows what a cretin he is.
She would not be "lessoned," so he says; and though she was always courteous with him, he complains that she acted the same with everyone, which he resents. And so he put an end to it—he bullied her into submissiveness...or it could mean he had her killed.
Oh, sir, she smiled, no doubt,
Whene'er I passed her; but who passed without
Much the same smile? This grew; I gave commands;
Then all smiles stopped together.
The Duke, at the least, crushed her spirit with his petty jealousy.
Then, in an instant, the Duke changes the subject—getting to the business of marrying again. He has been speaking with the emissary of his future bride's family. He speakes about the arrangements (the dowry). He is not one who knows affection. As they prepare to leave, he points to another piece of art, as if the Duchess was only that. It may also symbolize his undeniable power—and like Neptune—"taming" his wife:
Notice Neptune, though,
Taming a sea-horse, thought a rarity,
Which Claus of Innsbruck cast in bronze for me!
The narrator finds the Duke deficient—he does not want a meaningful relationship. He wants his way. He is selfish. His pride (a theme) and his need for control prohibit a true "emotional bond."