Consider the Victorian socio-political & historical context.
The Duke in "My Last Duchess" wants the ultimate trophy wife, as we would think of it today. He wants a beautiful wife who will give her attention to only him. She is to stand around like a work of art and reflect back on the Duke, demonstrating his good taste.
The Duke's view of women and wives is an extreme application of Victorian views of women. Society was extremely sexist in Browning's time, seeing women as practically being owned by their husbands. Women were not allowed to vote, and had little opportunity for social or economic advancement, other than marrying a husband of a higher status than themselves. Victorians, however, did not make a habit of killing wives who did not please their husbands.
Specifically, though, concerning the character of the Duchess, she is probably an ideal Victorian woman. She is beautiful, loyal, polite, and attentive to her husband. She is a kind, gracious hostess and admiring of nature. She is an ideal Victorian woman.
She just doesn't live up to the Duke's obsessive requirements. She doesn't limit her smiles to only him; she doesn't fawn over him because of his ancient name. She doesn't patronize him and feed his arrogance well enough.
She is an ideal woman, but she is not good enough for him. He is extreme, even for a Victorian.