1) We learn in this scene that Claudius believes that Ophelia's madness has sprung from the death of her father. He doesn't connect her death in any way with Hamlet, and his maltreatment of her may have in fact been the reason why she went crazy. This would help the actor play a Claudius that does not believe that Hamlet and Ophelia had a relationship. This could indicate that Claudius is unable to recognize love, therefore he may have no idea what it really is. Additionally, he is not really in love with Gertrude, rather, he sees her merely as a tool for political power and sexual gratification. We also learn that Claudius had Polonius's body buried in secret, however, it was in vain, for rumors of Polonius's death run rampant amongst the people. At one point, Claudius calls for his "Swissers" (guards), proving that he has become increasingly worried about his protection. This would give an actor the chance to demonstrate this. However, Claudius seems unafraid of Laertes. This could be because he never thought much of him in the first place, but he sees him as a potential tool in his plan to regain control and settle down Elsinore.
2) Claudius's objective does not become evident until Laertes enters. We discover that he intends to manipulate Laertes in order to have a ally that also wants to kill Hamlet. Laertes suspects that Claudius had something to do with his father's death, and Claudius promises Laertes that he will give him his kingdom if he is proven guilty or will help him get revenge on the person who did. Claudius knows that he is not at risk of losing his kingdom to Laertes, and it comforts him to know that there will be at least one other person who wishes to see the death of Hamlet. Granted, he doesn't yet know that Hamlet didn't make it to England, but he's bound to be grateful for Laertes's quest for revenge later on.
3) Claudius seems upset at Ophelia's madness, but only in context of that fact its another item to add to a growing list of events that could have bad political consequences. Claudius does not seem to fear Laertes at all, for he must know that Laertes is easy to manipulate. At this point, Claudius is sure that Hamlet is off to his certain death, however, he is still wary of his own personal safety.
4) Laertes enters the scene, convinced by the rumor that Claudius killed Polonius, and calls Claudius a "vile" king. His ill will toward the King calms down, however, as Claudius begins to question him about what he will do to avenge the death of his father. Laertes becomes convinced that the King will help him, demonstrating Claudius's power of persuasion. Claudius doesn't seem to care much for the personal well being of Ophelia, and he doesn't see Laertes as any sort of a threat, but as a potential tool to be used.
5) Claudius installs a sense of order in this scene, especially when Laertes enters with the mob of people. He calms Laertes down and makes some agreements with him, eventually satisfying him and promising to help him avenge the death of his father. It seems as thought Claudius resumes his sense of kingly power in this scene, and he again demonstrates his ability to solve an impending crisis.
6) Claudius is important in this scene because we see how Laertes will be intent on killing Hamlet once he finds out that is was the prince who killed his father. We also get the chance to see Claudius's crisis-handling skills in action, and they seem to be pretty effective. Claudius is a master manipulator. I didn't gain any insights on Hamlet.
7) I wonder how many people Claudius is going to try and set against Hamlet. Also, how is he going to tell Laertes that it was Hamlet who killed Polonius?
8) I know now that Claudius will manipulate people in order to ensure that he has allies against Hamlet. Claudius is unable to face threats on his own. He must have other people on his side if he wants to win.