Can you give me an example of Hamlet manipulating Gertrude to act to his benefit in Shakespeare's Hamlet?
Oh, Hamlet is an absolute master of manipulation! (If only he were master of action! Ha!) I think it best to begin with a lesser example and end with the most important in regards to Hamlet influencing Gertrude.
First, observe a few of Hamlet's first words to Gertrude in Act I specifically as a response to what Gertrude says:
Queen: Do not forever with thy vailed lids / Seek for thy noble father int he dust. / Thou kno'st 'tis common; all that lives must die, / Passing through nature to eternity.
Hamlet: Ay, madam, it is common.
The importance here is the word "common" and its meaning in Shakespeare's time. The word can mean both "universal" but also "vulgar." Many critics see this line as some of Hamlet's first public commentary on Gertrude's marriage to Claudius. Equally important is how this line is acted. If acted with enough contempt and sarcasm, the audience can be sure that Hamlet is trying to stir guilt in Gertrude.
The most important instance of Hamlet's manipulation occurs in the famous "bedroom scene" of Act III. I would argue that Hamlet is manipulating Gertrude beyond guilt here. Hamlet is hoping that Gertrude will feel actual dread at her wicked deed. I believe Hamlet secretly hopes Gertrude will renounce Claudius. I suppose Hamlet doesn't succeed in his goal in that regard. That being said, let's look at Hamlet's manipulation. (In reality, the entire scene should be reprinted here! Ha! But for time's sake, I will lift a few important parts.)
First comes Hamlet's taunts:
Hamlet: Mother, you have my father much offended. / ... / Come, come, you question with a wicked tongue. / ... / You are the Queen, your husband's brother's wife, / And, would it were not so, you are my mother.
Here Hamlet lets the accusations fly! Gertrude has severely "offended" Hamlet's dad, sins by everything she says and does, and has committed incest by marrying Claudius. Hamlet wishes she were not his mother, of course.
After the queen wonders "what have I done" to deserve these taunts, Hamlet goes on to accusations:
Hamlet: ... This was your husband. Look you now what follows. / Here is your husband, like a mildewed ear / Blasting his wholesome brother. Have you eyes? / ... / What devil was't / That thuys hath cozened you at hoodman-blind?
Hamlet is attempting here for Gertrude to see her vast villany. He wants her to think on her original husband, Hamlet's father, and compare him to the piece of trash she has now ventured into the taboo by marrying. He accuses her of not being able to see the great sin right in front of her nose. A sin that will damn her to hell.
What good would manipulation be if it didn't render results? Check out the result of Hamlet's manipulation. Quite effective, eh?
Queen: O Hamlet, speak no more. / Thou turn'st mine eyes into my very soul, / And there I see such black and grained spots / As will not leave their tinct. ... O, speak to me no more. / These words like daggers enter in my ears. No more, sweet Hamlet.
Now, it's important to note here that some critics think that Hamlet and Gertrude become incestuous in this scene (as evidenced by Mel Gibson's version, as one of many). If one wants to believe this interpretation, then Hamlet's manipulation goes further than above!
The entire bedroom scene is an example of Hamlet's manipulation of Gertrude. He is attempting to get information out of her, and does so through slightly nefarious ways. He knows that Polonius, I believe, is hiding in the room.
Additionally, the entire play is a contrived effort on Hamlet's part to force a confession from Gertrude by reliving what happened the day his father died. By showing his hand, he forces the King and Queen to bare their souls as well.