In the beginning of this scene, Hamlet’s mom, Gertrude, and his uncle, King Claudius, are inquiring of Hamlet’s behavior to Rosencrantz and Guildenstern. There is a concern about Hamlet’s supposed depression, but the two men claim they do not know what is bothering Hamlet. They do, however, inform the king that Hamlet seems very excited to see the play that is supposed to be performed that evening. This is something the king and queen find hopeful, and they are pleased to hear it. After dismissing everyone, Claudius and Polonius, Ophelia’s father, hide in the room to spy on Hamlet. After Hamlet enters, he delivers his famous “To be or not to be” soliloquy.
I would say that there is a feeling of concern that permeates this scene, but when looking at the character of Hamlet, the reader can see feelings of contemplation, depression, and self-doubt, as Hamlet is grappling with whether or not life is worth it.
For the most part, the overriding feeling expressed seems to be concern for Hamlet's seemingly confused state, which he himself calls an "antic" disposition. Claudius enquires of Rosencrantz and Guildenstern, Hamlet's friends, about his strange behavior. He seems to be worried by the prince's strange actions.
The two gentlemen inform him that Hamlet does seem withdrawn but that he does not want to discuss the matter. They correctly identify Hamlet's behavior as a "crafty madness" which he uses to stay aloof. This makes it difficult for them to sound him out because he is, as such, not very forthcoming. They do confess to Queen Gertrude, however, that they had been "well received" by the prince.
Lord Polonius later enters the conversation when Rosencrantz and Guildenstern mention Hamlet's pleasure on hearing of the arrival of a troupe of players. He confirms their report that Hamlet seemed quite happy about the news and that he had asked Polonius to speak to the king and queen about the players. The fact that Hamlet seems happy pleases all concerned.
The first part of this scene is, on the whole, a charade. It is ironic that all the speakers express so much concern for Hamlet when they are actually more concerned about their own positions. Queen Gertrude is probably the only one who has a genuine concern for her son, whilst Claudius is afraid of what Hamlet may do to him since he has usurped the throne from him by murdering the prince's father.
Claudius has hired Rosencrantz and Guildenstern to spy on Hamlet and will later use them to accompany Hamlet to England, where the prince is to be executed on terms of his devious plan. The two gentlemen are not so much interested in their friend as they are in the reward awaiting them for their service.
Polonius, on the other hand, is Claudius' servile and pompous slave. He slavishly obeys every command from the king. Furthermore, he is concerned about Hamlet's relationship with his daughter, Ophelia. Polonius is, therefore, not really so much concerned about Hamlet as he is about himself and protecting his position.
In the end, though, all this fake concern turns against all those involved, and they meet their untimely doom for their malice.