This is an excellent question, for the identity of speaker is central to understanding not only the relationship between the speaker and his wife, but also in better understanding his "last duchess," a woman who can no longer speak to defend herself.
The first thing the speaker tells us is very important in comprehending what has transpired in the Duke's home.
That's my last Duchess painted on the wall,
Looking as if she were alive.
The reader knows now that the woman in the picture is dead. "Last" duchess infers that she will not be the Duke's final duchess. As we read on, the speaker describes the painting to the person that is with him. We learn that the picture is always behind a curtain so that no one's eyes but the speaker's are able to look upon "the depth and passion of its earnest glance." (While it is grammatically correct, I am startled by the use of "its" as opposed to "her" glance, as he seems to objectify her.) It would appear that the Duke's visitor is the only other person to have had the privilege of looking on the deceased woman's likeness.
The reader discovers that the woman found happiness, but not just in her husband's gaze:
Her husband's presence only, called that spot
Of joy into the Duchess' cheek...
This comment demonstrates how unhappy the Duke is that he was not the center of her attention. She was an innocent, She knows that the painter's compliments were only his attempts to be courteous, but still they brought her "joy," and the speaker faults her for this:
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.
The reader can begin to question the character of the speaker. Who could possibly complain that his wife was too happy? It seems a ridiculous statement, but note what her "looks went everywhere." This is in direct contrast to the will of the man who just mentioned that she found pleasure beyond seeing just his face. He is insanely possessive; his covetousness is further evident as he hides her image behind a curtain.
The speaker did not love his wife but wanted to possess everything about her: that her pleasure be found only in him and that no one take her attention away from him. In that he would not allow others to see his wife's painting, we can only imagine how he felt when other men in public looked at his wife.
There is a tinge of unhealthy resentment and obsession in the speaker. He resented that another man would hand her a "bough" broken from a tree with cherries on it. Anyone's attention she politely appreciated (with no inappropriate intent on her part). The speaker is enraged because she thanked these others as if anything they could do, say or give her could compare to the immeasurable honor he bestowed upon her:
She thanked men,—good! but thanked
Somehow—I know not how—as if she ranked
My gift of a nine-hundred-years-old name
With anybody's gift.
The Duke is essentially point out that nothing compares to the honor he paid her by giving her is ancient family name. His attitude allows the reader to infer that he is unreasonable in his expectations...that her kindness should be shared with no one but him because she married into an age-old family. (This is just another piece of evidence to support the knowledge that he did not love his wife.)
The speaker goes on to say that he did not "lesson" (teach) her about his difficulties with her behavior—which began to disgust him: she either did too much or not enough. How much easier it would have been if she understand what he wanted. He refused to do so. (Another example of his lack of concern for her or the marriage—and his constant attention to "self.") For him, "stooping" to educate her was out of the question. ("Stooping" conveys his sense of superiority over her.)
...and I choose / Never to stoop.
His former duchess continued to smile at others, but wasn't she bestowing that same smile upon him—as if the others were his equal? His jealous nature has become impossible for the reader to ignore. His irrational and insane hatred of her is apparent. Then he points out that he let his anger be known to her:
I gave commands;
Then all smiles stopped together. There she stands
As if alive.
At this juncture of the poem, we can only wonder what has happened to his "last duchess." The Duke's bullying may have caused his beautiful and gracious wife to stop smiling completely, but the reader should ask how it really happened? Was it simply because he was so demanding and oppressive that she became sad? Or could it be that death caused this change? For without missing a beat, he immediately draws his companion's attention back to the painting, pointing out how alive she looks. This sudden shift in his conversation allows him to intentionally circumvent any question that might arise as to what actually happened to her.
There can be little doubt that his "last duchess" is dead, and there is good reason to believe that he was involved in her early, unexplained death.
As the poem moves to its conclusion, we note that the visitor has heard enough and jumps up. The Duke recognizes that his visitor is ready to leave—saying, "Oh! You're getting up?" He notes then that he will follow the man: "Fine then, we can both go down to see the rest of the group."
Will't please you rise? We'll meet
The company below, then.
The Duke makes all speed to keep up with the man who is, it appears, the representative of the Count. Apparently, he is there to see to the arrangements of marriage between the Count's daughter and the Duke. The Duke speaks of the woman's dowry: an advantageous marriage to the Count's daughter would bring money into the Duke's coffers. However, this deceitful and conniving man insists that his true interest is in the Count's daughter. We can also infer that if the first wife and her lovely disposition did not please him, the second probably will not either, but the Duke does not care because money is his desire, not another wife. Fortunately for the young woman in question, the Count's emissary will not be promoting this union—he sees the Duke as he truly is.
The reader realizes that the last duchess was a lovely and well-liked woman. While the Duke is trying to promote an inaccurate image of the former duchess, he manages instead to convey what a monster he is! The impact of Browning's choice of speaker allows the reader to understand how horrific this woman's life was. Had an admirer written this poem, we might imagine him to be an obsessed stalker. In that it is her husband, we recognize that this woman was a prisoner from the very beginning. She did not idolize her husband: she was too much her own person. His insane nature would have made it impossible for any woman to please him. Unknown to her, this sealed her fate from the moment they married. Instead of a duchess, she became a victim instead.