In "My Last Duchess," how did the duke respond to the duchess’s behavior at first?

At first, it seems that the duke responded to the duchess's behavior by taking note of everything that seemed to make her happy: a cherry tree branch, a white mule, a sunset, the duke himself.  The modesty or value of the gift made no difference to her; there was no distinction in the joy she felt, whether the gift was large or small, valuable or cheap.  Finally, the duke became offended that "she ranked / [His]...

At first, it seems that the duke responded to the duchess's behavior by taking note of everything that seemed to make her happy: a cherry tree branch, a white mule, a sunset, the duke himself.  The modesty or value of the gift made no difference to her; there was no distinction in the joy she felt, whether the gift was large or small, valuable or cheap.  Finally, the duke became offended that "she ranked / [His] gift of a nine-hundred-years-old name / With anybody’s gift."  In other words, she cherished his gifts, including the gifts of his status and wealth, only as much as she appreciated any other, smaller, gift, and this upset him.  He says,

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. 

What he means is that, even if he had the skill to make her understand (a skill he claims not to have), it would have required him to "stoop" or to lower himself.  He doesn't want to have to ask her to love him the best, to rank his gifts as the most worthy and valuable, i.e. not to put a bough of cherries on par with his name and title.  He isn't willing to lower himself in this way, and so he never speaks to her about it.  

Finally, he says that "[He] gave commands; / Then all smiles stopped together."  So, it sounds as though he had her killed so that he could begin again with a new duchess that would appreciate him most.  If he was unwilling to speak with her about what upset him, then he certainly wouldn't have "command[ed]" her on the subject.  But since her smiles stopped at this point, we can assume that this is how and when she died.