In Act 2, scene 1 we get to know Polonius a little better. Polonius is talking to a hired spy, Reynaldo, whom Polonius has hired to go to France in order to check up on Laertes and his reputation around town. Polonius thinks it would be a good plan to have Reynaldo put out little negative comments, "what forgeries you please" about Laertes in order to see how people respond to the comments. This seems rather short-sighted of Polonius, and Reynaldo even says, "that would dishonor him." Polonius isn't thinking about the fact that if Reynaldo casually says something negative about Laertes and people say they never see that bad behavior, the seed is still planted that Laertes does do the bad behavior. Polonius seems very concerned with reputation, but kind of foolish in his attempt to check up on Laertes and his reputation. We also realize the spying is something that Polonius has no problem with doing.
The rest of scene one is Ophelia's report to her father about Hamlet's very strange behavior. One thing the scene does is reveal that Ophelia has, in fact, done as her father asked and rejected Hamlet's attentions. It also reveals Hamlet's "crazy" act, and how he is going about that part of his plan. He seems to have been convincing to Ophelia.
Scene 2 reveals that Claudius and Gertrude have sent for two of Hamlet's friends, Rosencrantz and Guildenstern. We see that these two men seem to be more loyal to the throne than to their friend if they are willing to do as they are bid. We are at least suspicious of their intentions. We also learn that Claudius and Gertude are disturbed by Hamlet's crazy act, so again, the act must be convincing.
We also see Polonius's interactions with the King. Polonius clearly wants to be "right" about Hamlet and be favored by the King. He has a rather high opinion of himself, claiming "has there ever been a time when I have said, tis so when it proved otherwise?" Usually people who say things like that are brought down a peg or two eventually. Ultimately, this scene shows us again that reputation is important to Polonius.
When Hamlet appears at court we see first hand his "crazy" act. He is clearly making some barbed comments, but it seems that everyone present is left in the dark. The crazy act continues when Hamlet first speaks with his friends, Rosencrantz and Guildenstern. When they first talk, Hamlet seems pleased to see them, but his suspicions are quickly aroused and he loses his patience with them completely when he directly asks if they were sent for, and they don't immediately tell the truth. Hamlet writes them off as untrustworthy after the whole conversation is over.
By the end of Act 2, Hamlet seems to be surrounded by people he can't trust, but who are all desparate to figure out what is going on with him. He knows that he must move forward with his plans to assertation Claudius's guilt and avenge his father's death.