His Disney Wiki page is pretty good, actually: link
Here, he's considered the "secondary antagonist" (Hans obviously being the main antagonist).
Here is a section of his "trivia" from the Disney Wiki page:
While selfish, greedy and pompous, the Duke is not actually completely evil, since he cared for Anna when she returned, and he's very shocked when Hans told them Anna was dead. He is simply one of Hans' pawns, since he was lied to and manipulated. However, he did order his bodyguards to assassinate Elsa even without Hans' knowledge, which earned him his punishment of being cut off as a trading partner of Arendelle at the end of the movie.
The Duke is a red herring. At his first appearance, he reveals himself to be wanting to "exploit Arendelle's tradable goods" in a monologue. He follows with "Did I say that out loud?" This makes the viewer believe he is the villain, even Mehr so when he calls Elsa out for sorcery, and again when he sends his goons to kill the queen. However, the whole time, he is merely a cover-up for the real villain Hans.
Thought you'd find that interesting, and hopefully it's helped to answer your question.
Of course, some people might say that Hans is the only villain, and others might say that both characters are equally villainous. Personally I like the idea of calling Hans the primary villain and the Duke the secondary villain though.