What examples are there of Hamlet manipulating other characters to his benefit in the play?
Hamlet uses several characters for his own means:
1. Opehlia: Shakespeare hints at the fact that Hamlet might know that her father and the King and Queen have set her up to get information from him, or to at least be able to figure out why he is acting weird. So, when Hamlet meets Ophelia and she returns his "love tokens", his behavior is extremely erratic. How would this benefit him though? He is angry at the King and Queen, and wants to throw them off, confuse them, and vicariously express some of his anger at his mother's fickleness (hence the rant against women during that encounter).
2. Polonius: Hamlet once again acts very strangely around Polonius, probably because he knows that Polonius consults with the King and Queen. It's another way to confuse them all, and throw them off of his real quest: figuring out who murdered his father. If everyone thinks he's crazy, then they won't suspect his true mission, and he can get closer at the truth.
3. The Players: Hamlet has them enact a play of his choosing, that closely resembles the suspected murder scene of his father. He does this to gage his uncle's reaction, in order to determine his uncle's guilt or innocence.
These are just a few examples of characters that Hamlet manipulates in order to achieve his own ends, which is to ultimately figure out what happened to his father, and express some of his pent-up repugnance with his mother's actions.