In Hamlet why is Hamlet so happy when King Claudius becomes upset by the play?
The reaction that Claudius has in response to the play seems to indicate that he is indeed guilty of murdering his brother. Hamlet set up the play to mimic almost exactly the suspected murder; he did that so that he could watch the king's reaction, to see if it gives anything away. Hamlet, visited by his father's ghost at the beginning of the play, doesn't want to move forward with revenge until he is reassured of his uncle's guilt. A ghost was the one to tell him after all; perhaps it was an evil spirit meant to trick him into murdering his innocent uncle. So, Hamlet wants proof of his uncle's guilt before he goes murdering him.
The play is a chance to stage a scene much like the actual murder, and then watch the king's reaction. Imagine your worst deeds acted out in front of you on a stage; you would probably be pretty upset. If the king wasn't guilty, the scene wouldn't have upset him so much--it would just be a good, entertaining story. But, the king gets super upset--this indicates he recognizes the scene, and feels that someone is conspiring against him somehow.
This makes Hamlet happy, because now he is assured of the king's guilt, and can finally move forward to enact revenge, without any sense of guilt or regret. Killing a man guilty of killing your father is revenge, not murder. Hamlet's wondering is at an end. I hope that those thoughts helped; good luck!