Character Analysis
We first see the Cardinal as he walks in with his siblings, and boy has Antonio got some choice words to describe him:
He is a melancholy churchman.
[…]
Where his is jealous of any man he lays worse plots
for them than ever was imposed on Hercules, for he strews
in his sway flatterers, panders, intelligencers, atheists, and a
thousand such political monsters. (1.1.153-58)
The Cardinal comes off as the evil-doing mirror image of Ferdinand: he's smooth, calm, and emotionally detached. Where Ferdinand is passionate, turbulent, and frequently linked with images of fire, the Cardinal seems totally devoid of emotion, which both makes him an effective villain, but also a kind of confusing one for this play.
In fact, the Cardinal is probably the only character in this play who isn't motivated by passion, and the fact that he's a smooth criminal actually makes his involvement with Ferdinand's vendetta against the Duchess difficult to understand.
Machiavelli and the Cardinal: Woulda Been Bros in Another Life
At this time, there was a lot of popular anxiety over the decay of traditional Christian ethics, and people largely blamed a guy named Niccolo Machiavelli, a 15th-century Italian political philosopher whose basic message was "screw morals; do what you gotta do to stay in power." Although Ferdinand's all down with the plotting, he's too ruled by his passions to be a true Machiavellian villain.
The Cardinal, though, has got all of Machiavelli's favorite personality traits: he's manipulative, ruthless, creative (poisoned bible, anyone?), has a gift for the wheeling-and-dealing of treacherous secrets, and most of all he's able to do it all with a level head, unperturbed by passion or remorse.
As a "melancholy churchman," the Cardinal in particular illustrates the division of power and good old Christian morality, which people figured was going the way of VHS and the floppy disk. One of Machiavelli's main messages was that you have to look like you've got religion, even though you can't let yourself be tied down by all of the pesky moral stuff that could get in the way of your achievement of power.
The Cardinal has all of the political power that comes with his religious position—he eventually pulls some strings with the Pope to both seize the Duchess's lands and get her banished from Ancona—and none of the moral obligations that traditionally bound priests. It's a pretty sweet deal for a cold-blooded megalomaniac like the Cardinal, and a pretty bad deal for… basically everybody else.
Or Would They Have?
It's difficult, however, to suss out exactly what the Cardinal's endgame is.
Ferdinand comes off more as a dramatic villain than a political one because basically everything he does he does to gratify his (really deep-seated, really messed up) psychological and emotional needs. The Cardinal, though, is consistently cool-headed, and can plot to kill his own sister without so much as batting an eye. As far as we can tell, he has no emotional needs—at the moment when he decides he wants to get rid of his mistress Julia, he says "Yond's my lingering consumption: / I am weary of her and by any means / Would be quit off" (5.2.220-23).
Point being? This man is stone-cold.
That being the case, his involvement with the Duchess's murder is actually kind of weird: the Cardinal is clearly into power—the achievement of it, and the twisted machinations that go into maintaining it—but it's not clear what that has to do with killing his sister.
With Ferdinand, we're given a lot of material (albeit confusing material) dealing with his feelings towards the Duchess. With the Cardinal, we have to do a lot more guessing. Upon hearing of the Duchess's newborn child, the Cardinal says, "Shall our blood, / The royal blood of Aragon and Castile / Be thus attainted?" (2.5.21-23), so it does look like he's disturbed by the idea of a corrupt lineage, but this doesn't explain his initial insistence that the Duchess not remarry. There isn't a clear political motive in killing her, and even though he could conceivably be after her wealth, that motive is as suspect in the Cardinal's case as it is in Ferdinand's.
What's your take?
Probably Not Actually Cut Out For Machiavellian Villain of the Year
While the Cardinal comes off as a pretty complete package with regards to Machiavellian Perfection, he fails in two important ways:
(1) As aforementioned, there is no clear political or power-related motive to the Cardinal's pursuit of the Duchess's destruction. If he were, as some people assume, simply out for the Duchess's wealth, he would have had her done away with ages ago. Why the brothers are out for the Duchess is a tough question, and almost all of the information we get on that question is attributed to Ferdinand, who does all of the emotional and psychological heavy-lifting.
This leaves us having to conclude that the Cardinal is doing evil for the sake of evil, even when that ultimately comes at the cost of his power (and life). Machiavelli would be shaking his head in disapproval at this—he's down with lying and cheating, but really only in pursuit of power.
(2) He loses control of the situation. A true Machiavellian villain is characterized as a flawless puppet-master, pulling the strings from afar of all of the small-fry villains and getting them to do his work for him. The Cardinal does a bang-up job of this for the majority of the play, but he ends up dropping the ball big-time when he fails to realize that one of his tools, Bosola, has had a change of heart and intends to kill the Cardinal himself. The Cardinal's plan to cover up Julia's death ends up running afoul of his plan to have Antonio killed, and the Cardinal ends up getting murdered while his own courtiers listen.
As good as the Cardinal is at scheming and plotting, we're thinking that in the end he would get voted off the island on Survivor: Machiavellian Villain Edition. Or, rather, he'd be betrayed by the other contestants he thought he was manipulating and then forced to throw himself into shark-infested waters.