The Duke actually begins to express his arrogance, though subtly, in the first few lines of the monologue:

. . . for never read
Strangers like you that pictured countenance,
The depth and passion of its earnest glance,
But to myself they turned . . .
And seemed as they would ask me if they durst,
How such a glance came there.

At least after we have read further, in looking back on these lines we can see he's already implying that there was something wrong with that "earnest glance." He then tells his auditor that

She had a hear—-how shall I say—
Too soon made glad, too easily impressed

At this point, his judgmental nature emerges without any subtlety, the complaint being, if we may paraphrase him, that she seemed to like everyone, not just the duke. She was not sufficiently awed by his "gift of a 900-year old name." Now the floodgates of his criticism are open, and he begins to elaborate. She smiled at others just as she did to him; he did not like this, and he states (almost with pride) that he then "gave commands" and her smiles stopped altogether.

The afterthought of his dissection of his relationship with the last duchess is his casual reference to the figure of Neptune and the sea-horse "which Claus of Innsbruck cast in bronze for me." The impression is that the painting of his former wife is just another artwork, no more significant than the bronze casting.

Robert Browning's first-person poem "My Last Duchess" is masterfully loaded with arrogance. The concept of the poem alone reeks of arrogance, as the narrator is bragging proudly about the painting of his "last" duchess, who he had killed.

Here are some of the clearest examples of the narrator's arrogance:

In the fifth line of the poem, the narrator asks his visitor, "Will't please you sit and look at her?" After bragging about the painter for worked busily on the portrait, he is offering the man a seat, expecting anyone would just be delighted to sit down and take in his beautiful painting of his dead duchess.

In line 23, the narrator describes how the duchess was too "impressed" by everyone, and was too nice. "Too easily impressed; she liked whate’er / She looked on, and her looks went everywhere." This suggests that she was sweet and admired by many for her kindness and beauty. This is where it starts to become clear that the narrator killed his innocent, young wife because he couldn't tolerate others appreciating what was supposed to be his.

Later he adds, "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 narrator feels like his wife was too grateful to others, and therefore disgraced his family's royal name.

Further down, he describes how he could have simply just explained to the dutchess that he didn't like the way she smiled so much at everyone and the other various issues he had with her, but he refused to do so because it would be beneath him: "—E’en then would be some stooping; and I choose / Never to stoop."

"Oh sir, she smiled, no doubt, / Whene'er I passed her; but who passed without / Much the same smile?" The Duke does not feel like she treats him as though he is better than everyone else, which, in his mind, of course, he is.

So he chooses to have her killed:

"This grew; I gave commands;
Then all smiles stopped together. There she stands
As if alive."

At the end of the poem, he simply moves on to show off another precious piece of artwork.

What words or lines especially convey the speakers arrogance in Browning's  "My Last Duchess"?

"She thanked men,—good! but thanked Somehow—I know not how—as if she ranked My gift of a nine-hundred-years-old nameWith anybody’s gift."

"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."     

It is clear from the first quote that the Duke valued his nine-hundred-years-old name, his "gift" to the Dutchess, as something  that placed him above other men ... or at least should have placed him among other men in here sight.   The second quote shows his supreme arrogance and possessiveness. Yes, she smiled when he passed; but all others who passed, who had not given her his "name," received the same smile.  This clearly was a violation of his "specialness."  If he didn't get the "special" smile, then they would all cease.

There's arrogance every place you look.

What words or lines especially convey the speakers arrogance in Browning's  "My Last Duchess"?

"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;" (9-12)

These lines reveal the speaker's sense of self importance. When discussing the painting, he mentions that others have come to him as an authority.