Certainly, Act III of Hamlet is central to the plot of Shakespeare's most renowned play that contains a most intriguing and transcendent main character.
***In Act III, Scene 1, a theme of treachery is introduced with the report from Rosencrantz and Guildenstern that Hamlet has arranged for the performance of a play that night. The former friends of Hamlet are instructed by Claudius to keep Hamlet distracted with this play and they leave. Then, Claudius tells Gertrude to leave as he and Polonius--"lawful spies"--will hide behind a curtain in order to judge whether Hamlet is insane or simply in love.
Also, in this scene, with the famous "to be or not to be" soliloquy of Hamlet, the theme of existential meaning appears as Hamlet wonders whether it were better to end his suffering and if the only reason people do not is their fear of the hereafter. In this despondent frame of mind, Hamlet then speaks with Ophelia and in his concern for her, he tries to push her away desparaging marriage and suggesting that she go to a nunnery. Ophelia misinterprets his misanthropy for insult, and, thus the theme of Hamlet's madness begins.
After hearing the conversation of Hamlet and Ophelia, Polonius suggests to King Claudius to have Hamlet come to his mother and he will hide behind the curtain; if Gertrude cannot find out what is wrong, then Polonius treacherously suggests that the king send Hamlet to England or lock him up wherever is the best place. Claudius agrees,
Madness in great ones must not unwatch'd go (3.1.197)
***In Scene 2, Hamlet's dissembling is a theme as he has the players reenact the death scene of his father, King Hamlet, in order to watch the reaction of Claudius and of his mother, Queen Gertrude. After the play, the theme of retaliation is foreshadowed as Ophelia asks Hamlet "What means this, my lord?" as the dumshow proceeds, and Hamlet replies "mischief." then he asks his mother what she makes of the show. Gertrude replies that “The lady doth protest too much, methinks,” suggesting that the queen does not think remarriage for a widow unthinkable as does the actor. When Rosencrantz and Guildenstein appear to escort Hamlet to his mother's quarters, they report that Claudius is in distemper and suggest the importance of preserving the king's health. This speech of theirs seals Hamlet's retaliation against them for their treacherous loyalty to his father's murderer.
***In Scene 3, King Claudius conspires further with Rosencrantz and Guildenstern and Polonius. Then, his soliloquy reveals his guilt about killing his brother, and this guilt becomes the theme of the scene as Hamlet overhears him, but cannot kill him while in prayer lest Claudius become a martyr.
***In Scene 4, the themes of madness and treachery continue as Hamlet talks with his mother and insists that he is not insane. He pleads with his mother, to look at her innermost self, and her marriage is
Such an act
That blurs the grace and blush of modesty;
Calls virtue hypocrite; takes off the rose
From the fair forehead of an innocent love,
And sets a blister there; makes marriage vows
As false as dicers' oaths (3.4.45-50)
Hamlet's speech also echoes the recurring theme of appearance vs. reality in the play. Still, according to critic Harold Bloom, the existential theme of the soliloquy "to be or not to be" of Scene 1 is intrinsic to the play, declaring, "It is the foundation for nearly everything he will say in Act V."