Hamlet does not have a soliloquy in Act I, scene 1, but you might want to check out the free Shakespeare Quotes section link below for more information. The link should take you directly to Hamlet quotes.
Hamlet's soliloquy in Act I, scene 2 is about how much he wants to die. He's crying out with the pain of his existence and wishes he could kill himself. Hamlet also describes the reasons he feels this way. These provide much of the dramatic purpose of the speech.
He feels this way because of the sight before him. He sees his mother married to his uncle, the brother of his father, when his father and King had died only recently. He sees Claudius as inferior in all ways to his dead father and is doubly sickened because his mother hangs her affection on Claudius so strongly.