The character of the Duchess is described in the second half of the poem. As the Duke tells his acquaintance about his former wife, it becomes clear that the Duchess was a bit of a flirt. At least that is how the Duke interprets it. You have to remember that it is the Duke telling the poem, and therefore the point of view is skewed to his opinions and thoughts. He saw the Duchess as a flirt and a tease. He says that the Duchess thanked all men equally and smiled the same at all men as well. He says that she looked on other men's gifts the same way that she looked at his gifts. In his opinion, the Duchess seems to be on the prowl for whatever man may keep her attention at that moment. He also says that she is far too easily amused. Which the Duke interprets to mean that she doesn't commit to any one thing for very long . . . including him.
". . . too soon made glad,Too easily impressed; she liked whate’erShe looked on, and her looks went everywhere.""She thanked men—good! but thankedSomehow—I know not how—as if she rankedMy 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 withoutMuch the same smile?"
"This grew; I gave commands;Then all smiles stopped together."