The Duke describes the last Duchess as if she were wanton with her attention, inadequately class conscious and overly friendly. As rowens says, his primary complaint is that she does not treat him with more reverance or favour than she does anyone else. She loves everyone and everything: "she had a heart too soon made glad, too easily impressed". She enjoys life and people, and does not differentiate between nature and art, high or low class, men in general and her husband with the 900 year old name.
She is not a coarse woman--she blushes easily (too easily, according to the Duke, and too frequently as well). She smiles at everyone (friendly, but undifferentiatingly so), and she thanks everyone (this makes her too free with her gratitude). As far as the Duke is concerned, she should only be thankful to him for giving her his name (and title).According to the Duke, she is oblivious to her faults, and does not correct herself (he doesn't tell her he sees anything wrong with her behaviour because that would be "stooping" and the Duke never stoops.
So her three greatest faults are that she is uncommonly friendly (which makes her common in the Duke's eyes), she is insufficiently grateful and subservient to the Duke, her husband and, finally, that having made these grave errors she does not see them and correct them on her own (thus putting the Duke in the uncomfortable position of feeling he must tutor his wife, which of course he cannot do).
From his description of his last duchess, you would think the duke would adore her. She is sweet and kind and easy to please. However, quite the contrary is true of the duke. The following lines describe the duchess' personality as reported by the duke. I have added commentary in bold:
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 duke says that she is a happy person and that even small things make her happy. Any small compliment would call a blush to her cheek. BUT according to the duke, this joy in small things was a fault.
Sir, 'twas all one! My favour at her breast,
The dropping of the daylight in the West,
The bough of cherries some officious fool
Broke in the orchard for her, the white mule
She rode with round the terrace—all and each
Would draw from her alike the approving speech,
Or blush, at least....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 upset that everything pleases the duchess. She is too innocent and too generous with her approval and too easily impressed. He later says that her holding any kindness in equal standing with his affection and his gifts "disgusts" him, though he refuses to "stoop" to tell the duchess why he is upset. She should know without his telling her. The duke thinks that the duchess did not show him the proper amount of gratitude for his granting her his name. He fails to see the beauty in her sweetness.