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Julia is the wife of the aging courtier Castruchio and the mistress of the Cardinal. With regards to plot, Julia doesn't do way too much until the very end, at which point she tries to finagle secrets out of the Cardinal and gets murdered via poisoned Bible for her troubles. In larger terms, Julia both illuminates the character of the Cardinal and acts as a foil for the Duchess.
Julia's relationship with the Cardinal is interesting for two reasons:
The first is what it reveals about the Cardinal himself. In short, this guy is cold. He has no scruples about having an affair with one of his courtier's wives, and is just as blasé about having her killed once he's grown tired of her. Gross.
Secondly, there's the matter of Julia's death. While the Cardinal opens the movements leading up to her death by saying that "by any means / [I] would be quit off" (5.2.222-23), it's not clear that that's why he kills her. The Cardinal is a crafty guy, and he could have gotten rid of Julia in a thousand other ways that didn't leave a body. With that in mind, let's look at that scene a little.
It opens with Julia asking the Cardinal what's on his mind, and he replies that he can't tell her because he can't trust her to keep his secrets, saying,
[…] think what danger 'tis
To receive a prince's secrets: they that do
Had need have their breasts hooped with adamant
To contain them. (5.2.251-54)
Julia insists, and eventually the Cardinal tells her that he's had his sister and her kids executed, and finishes of saying, "Think you, your / Bosom will be a grave dark and obscure enough /For such a secret?" (5.2.263-65).
While we've got a lot of information on the other characters' views on princes and aristocrats, the Cardinal's personal twist is an interesting one: a prince, here, isn't so much the product of lineage and blood as he is of his "adamant-hooped heart."
In that case, Julia's biggest misstep in this scene isn't that she overestimates the Cardinal's regard for her (although she definitely does that), but that, politically, she's trying to run with the big dogs when she herself isn't "a grave dark and obscure enough" to handle the intrigue and machinations of that sphere.
The Cardinal never would have let Julia live knowing his secret, whether or not she proved able to keep it, so Julia's own admission that she can't keep quiet about the Duchess's murder probably isn't what makes the Cardinal pull the trigger on killing her. For our part, we think Julia was dead the moment she started insisting she could keep such a secret; her murder is punishment for her bold attempt to become the Cardinal's aristocratic equal.
Even though these two characters never share a scene, Julia's presence in the play definitely shapes how you view the Duchess. For more on this, go check out the Foil section of "Character Roles."