How do Hamlet, Polonius, and Claudius use disguise to subvert their true natures and purposes and thereby achieve power? My main topic here is "Appearance Vs. Reality."
The themes of disguise and "appearance vs. reality" are central to the play; namely in that certain characters "act" differently in order to fool others. Claudius, Hamlet, and Polonius use disguise as a means to achieve some end, and you could argue that the "end" is power for each.
Claudius appears or disguises himself as the new, dutiful king, loyal to the legacy of his brother and genuinely in love with Gertrude. This "act" or disguise is a mask for what he really is: a murderer who pretends to be the dutiful king in order to achieve the power of being king. Claudius is acutely aware of his duplicity. Note in Act 3, Scene 3, the scene ends with Claudius saying, "My words fly up, my thoughts remain below: / Words without thoughts never to heaven go." Claudius appears to be in a penitent, sacred state; so, Hamlet decides not to kill him yet. This was fortunate, temporarily for Claudius, but in general, he tries to appear this way, except when the audience hears his dialogues with Polonius and Laertes in trying to secure his power from Hamlet.
Hamlet's entire strategy is about deception and disguise. Although Polonius is often mistaken about things, he rightly points out that Hamlet's madness is a disguise for some other purpose. "Though this be madness, yet / there is a method in't." (II.ii.216-17) Hamlet's use of appearing melancholy, then mad is all a means to keep others guessing and unsure about what Hamlet is up to.
Polonius appears to befriend Hamlet (Act II, Scene 2), but again, his intention is to spy on Hamlet and find out the "method to his madness." Polonius says he has the ability to figure all of this out, but he's often mistaken in his conclusions. His efforts are to appease Claudius and secure his position as a kind of right-hand-man to the king.
Each character "acts" a certain way in order to achieve some position. Claudius acts like a gracious king in order to be the king: for power. He also acts like a friend to Hamlet but that is to protect himself from Hamlet and perhaps get rid of him. Polonius "acts" in different ways, but all for the selfish reasons of being in a high government position with Claudius as leader: a powerful position. Hamlet "acts" mad (crazy) to keep everyone unsure of his method (his plan for revenge): a position that gives him the power and/or space to carry out his revenge.
This is an interesting topic in that it shows how this play is really about "acting." These characters are often "not themselves" in the sense that they are always acting in different ways. This is fitting considering that there is a play within this play ("The Mousetrap"), since the characters also behave like actors utilizing different roles in order to confuse the others.
Claudius attempts to disguise himself as a father-figure to Hamlet, as someone who cares for and will look out for Hamlet's best interests, when he is really just trying to protect himself and his own power. He says to Hamlet,
We pray you, throw to earth
This unprevailing woe, and think of us
As of a father. For let the world take note
You are the most immediate to our throne
And with no less nobility of love
Than that which dearest father bears his son
Do I impart toward you. For your intent
In going back to school in Wittenberg,
It is most retrograde to our desire. (1.2.106-114)
First, Claudius tells Hamlet to come out of mourning for his father's death, and Hamlet begins to think of Claudius as a father. He says that Hamlet is the person closest to his throne (truly, one might have expected Hamlet to inherit the throne when his father died), and he tells Hamlet that he loves him as a father does his son. He will not let Hamlet return to Wittenberg. Claudius implies that he wants Hamlet to stay because he cares about him, but—given the events that will occur—it seems more likely that Claudius just wants to keep an eye on Hamlet. Have you ever heard the expression, "Keep your friends close and your enemies closer"? If Claudius keeps Hamlet, the person with the most claim to his throne, close by, he can know what Hamlet is up to.
In Act 2, Scene 2, Polonius also tries to disguise himself as Hamlet's friend, in order to gain information from him, information that would raise Polonius's standing with the king. Polonius thinks the cause of Hamlet's madness is his love for Ophelia, a love that Polonius made her reject, and so he tries to sound out the prince. Verifying the source of Hamlet's madness would increase Polonius's power, but Hamlet refuses to answer his questions in any kind of a helpful way and thwarts the old man's attempts.
Hamlet obviously uses madness as a disguise in order to allow himself to verify his uncle's guilt and then avenge his father's murder; this disguise, he hopes, would confer some power on him because it might make his uncle underestimate him. However, perhaps as a result of his own inability to act in a timely manner, Hamlet is unable to survive the excision of the rot in the Danish court.