Courtiers, Antonio says, aren't just there for decoration. Although princes like Ferdinand surround themselves with useless sycophants, Antonio wants to be a courtier more like those found in the French court; the kind that actually provide useful advice to their princes, instead of just simpering and fawning.
I am making my will, as 'tis fit princes should […] (1.1.368)
The Duchess says this right before she proposes to Antonio. At the moment she says it this line sound like, "I'm a ruler, so it's important that I write my legal will," but once it's clear that she intends to woo Antonio, it can also sound like "I really want to marry Antonio, and as a prince it's right that I make my desires (my will) a reality." Which is cool, because it makes it sound like it's her duty to marry Antonio, because, hey, she's the boss, and that's how it works.
This is flesh and blood, sir,
'Tis not the figure cut in alabaster
Kneels at my husband's tomb. (1.1.445-47)
The Duchess says this during her wooing scene in Act 1, trying to persuade Antonio that she's a Real Woman, not just his distant ice-queen aristocratic boss. In this line, she's specifically fighting against the belief that woman can only marry once, and that a truly chaste woman is bound to honor her dead husband until her own death.
Think about this, though: the Duchess is always identified by the title she received from her dead husband, and all of her political power—"I am Duchess of Malfi still"—comes from him. What identity do you think the Duchess ultimately embraces more, that of the imperious widow of dead aristocrat, or of the wife of a living, un-aristocratic man?
I have heard you say that the French
courtiers wear their hats on 'fore the King.
Why should we not bring up that fashion?
'Tis ceremony more than duty that consists
In the removing of a piece of felt.
Be you the example to the rest o'th'court,
Put on your hat first. (2.1.119-28)
The Duchess figures that court protocol is really more like guidelines than actual rules. She wants to rule her court according to what makes sense to her, not according to weird rules that don't seem have any point. Why should people have to take off their hats in front of the king? Why shouldn't she marry who she wants?
Duchess: They have done wisely.
This puts me in mind of death: physicians thus,
With their hands full of money, use to give o'er
Antonio: Right the fashion of the world:
From decayed fortunes every flatterer shrinks,
Men cease to build where the foundation sinks. (3.5.7-11)
Now that things have started to really go south for the Duchess, most of her servants have abandoned her as she and Antonio flee her court. While it may seem kind of ungrateful for the servants to have left her flying solo, the Duchess herself understands their decision. As a fallen prince, she's no longer in a reciprocal relationship with them. She's ceased to function as their "foundation," so there's no reason for them to stick around.
In my last will I have not much to give:
A many hungry guests have fed upon me.
Thine will be a poor reversion. (4.2.189-92)
To Cariola's credit, she sticks with the Duchess until the very end. As she heads to her execution, the Duchess laments that she isn't able to reciprocate Cariola's loyalty with the kind of princely pay-off typically ensured to the "hungry guests" of a ruler. Take note, also, of the Duchess's use of the word "will" here—she's used it before (see Quote #2), and here again it retains its double sense of "legal document adjudicating property" and "personal desire." How would you describe the status of the Duchess's "will" at this point in the play?
[…] as we observe in tragedies
That a good actor many times is cursed
For playing a villain's part—I hate thee for't;
And for my sake say, thou hadst done much ill, well. (4.2.278-81)
In a moment that shocks even the jaded Bosola, Ferdinand turns on Bosola after her learns that the Duchess has been executed, per his own orders. He acknowledges that Bosola did exactly what he wanted him to do (done much ill, well), but says that Bosola actually should have defended the Duchess from him, and disobeyed him. Ferdinand has been all about Bosola carrying out his duties to him as minion and spy, but now that the Duchess is finally dead, he tells Bosola that he should have been an "honest man" instead of caring so much about his "duty" to be an intelligencer.
Let me quicken your memory, for I perceive
You are falling into ingratitude. (4.2.282-83)
Bosola flounders as he realizes that Ferdinand is betraying him. He's been thinking that he and Ferdinand are on the same page with this whole intelligencer-prince thing. Bosola is Ferdinand's employee, and the "ingratitude" he speaks of here is the same "ingratitude" he himself feared falling into if he refused Ferdinand's original offer of employment (go check out Morality and Ethics Quote #2 to see that moment in detail). Bosola's been doing all of this terrible stuff because he's been under the impression that Ferdinand will reward his loyalty, only to find out that Ferdinand has absolutely no intention of doing so.
Let me know
Wherefore I should be thus neglected. Sir,
I served your tyranny, and rather strove
To satisfy yourself than all the world;
And though I loathed the evil yet I loved
You, that did counsel it, and rather sought
To appear a true servant than an honest man. (4.2.317-23)
This is the moment where Bosola realizes the whopper of a mistake he's made in pledging his service to Ferdinand. He's known since the very beginning that Ferdinand was a bad guy—he doesn't even like this dude—but devotedly worked as his spy because he thought that Ferdinand would honor the traditional terms of the client-patron relationship and he'd get his in the end. The only way that Bosola was able to justify the fact that he was spying and murdering was to tell himself that he was doing it because he's a "true servant," only to discover in the end that Ferdinand isn't a true master.
I'll join with thee [Antonio] in a most just revenge:
The weakest arm is strong enough that strikes
With the swords of justice. (5.2.335-37)
This is Bosola's last-ditch attempt at duty relationships: he tried being the Cardinal's servant, he tried being Ferdinand's servant, and none of it's working. Having realized that he can't work for the bad guys anymore (or, for that matter, the one good guy, now that he's executed the Duchess), Bosola pledges to serve justice by avenging the Duchess, hoping that if he can just choose the right person or principle to serve, he can turn things around.