In Act 3, Scene 1, Claudius asks Rosencrantz and Guildenstern what they were able to learn about the reasons for Hamlet's odd behavior. Rosencrantz and Guildenstern reveal very little. They note that Hamlet warmed to the idea of the players performing and Claudius tells them to keep Hamlet's attentions on the players, to keep him in good spirits. (Ironically, Hamlet will later use the play to expose Claudius' crime.)
Claudius, Polonius, and Gertrude discuss other ways they might figure out Hamlet's mental state. Claudius and Polonius decide to hide while Hamlet talks to Ophelia. During their conversation, Hamlet chastises Ophelia, says he'd never loved her, and denounces marriage. Ophelia is hurt. Claudius and Polonius still suspect there is something more to Hamlet's odd behavior than any love-sickness he may have with Ophelia.
Claudius determines to send Hamlet to England to get his mind off of his troubles. However, Claudius later reveals that he sends Hamlet to England for his own protection. Polonius asks for one last effort to discover Hamlet's thoughts by hiding while he (Hamlet) talks to Gertrude.
In Act 3, Scene 1, Hamlet delivers the famous "to be or not to be" soliloquy where Hamlet weighs the pros and cons of life and death. As the play goes on, it becomes clear that Hamlet does choose life ("to be") and that his continuing reason to go on is to carry out his revenge.
In this scene, we see many of the characters struggling to figure Hamlet out. Complicating matters, many characters are not honest with each other. Claudius shares his motives with Polonius, but not Rosencrantz and Guildenstern. Hamlet lies to Ophelia. Rosencrantz and Guildenstern are initially dishonest with Hamlet and later reveal little to Claudius. Although Hamlet is hesitant and over-thinks things, it is his apparent "madness" and strange behavior that puts everyone on edge. So, while Hamlet continues to manipulate others, those others continue to try to discern what Hamlet is up to. Everyone is second-guessing everyone else's motives and honesty. Although unaware of Hamlet's actual intentions, Claudius is intelligently skeptical of Hamlet's motives:
It shall be so.
Madness in great ones must not unwatch'd go. (III.i.197-98)
Aside from Hamlet's famous speech on the merits of life and death, this scene is also about the many schemes characters undertake in order to discover other characters' motives.