The Duke is used to being catered to and almost worshipped for who he is and for his very old and respected family name. He speaks to the Count's representative in an authoratative, almost condescending tone, and hints that he will not tolerate anyone who does not treat him with the respect he feels he deserves. He speaks of the look on his last duchess' face:
Sir, 'twas not
Her husband's presence only, called that spot Of joy into the Duchess' cheek; ...She had
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.
This quote indicates that he perhaps thought she was cheating on him, or that she found joy in many things--simple things in life--which didn't deserve the same joyful look she gave him. He demands to be treated in a special way.
Somehow—I know not how—as if she ranked
My gift of a nine-hundred-years-old name
With anybody's gift. Who'd stoop to blame
This sort of trifling? Even had you skill
In speech—(which I have not)—to make your will Quite clear to such an one, and say, "Just this Or that in you disgusts me; here you miss, Or there exceed the mark"—and if she let Herself be lessoned so, nor plainly set
Her wits to yours, forsooth, and made excuse, —E'en then would be some stooping; and I choose Never to stoop.
He can not understand how she treated everyone as she treated him...her husband with an old and respected family name. How could she be so dense? He says, "Of course, I could have taught her; told her how she offended me, but that would be stooping beneath my stature." The Duke expects others to know how he expects to be treated, and he doesn't care enough about them to correct mistakes or direct them so that they could behave more to his liking. He just..."stops smiles altogether."
Notice Neptune, though,
Taming a sea-horse, thought a rarity,
Which Claus of Innsbruck cast in bronze for me!
The last lines in the poem follow his intent to marry the Count's fair daughter. All of this poem and the discussion of his last duchess' mistakes are fair warning for the next duchess--she had better act accordingly or her smiles will be stopped altogether, also. The Duke is an unforgiving, intolerant, and belligerent man. He bullies others into doing his bidding. He treats people as objects to own and admire (this is why he mentions the rare statue of Neptune cast in bronze just for him by Claus of Innsbruck)--remember that his last duchess is behind the curtain, standing "as if alive" (he had her murdered?), only for his eyes to see. He shows it to the Count's representative to open the conversation about the next duchess and his expectations for her.
If you were the Count's representative, would you recommend allowing the girl to become the Duke's next Duchess? Why or why not?