The ghost of Hamlet's father returns and instructs Hamlet not to "let thy soul contrive against thy mother. . . Leave her to heaven and to those thorns that in her bosom lodge to prick and sting her" (I.5). Clearly, Hamlet is unable to follow this direction from the ghost. Through the course of the play, trace Hamlet's underlying preoccupation with his mother, and show how this diverts him from his original goal of seeking vengeance. You should consider Hamlet's opinion of his mother at the onset of the play, and how this attitude develops throughout the play.

Hamlet's attitude to his mother from his first appearance seems one of quiet disdain. In act 1, scene 2, he somewhat coldly addresses her and seems to fixate on her statement questioning his ongoing grieving. She has asked:

If it be,/Why seems it so particular with thee?

You might ask how much of his manner is openly disdainful of Gertrude herself and how much is just Hamlet's ongoing grief limiting his ability to reach out to his mother. Yet he clearly is furious at her for marrying Claudius. Also, observe that Gertrude addresses him with the familiar "thou" but he uses the more formal "you" to her. Pronoun usage in Shakespeare's time is an interesting issue, especially since the familiar forms were already beginning to die out in general conversation. You might look up some information about the nuances of these pronouns in his characters' speech.
In act 3, scene 2, before and during the play-within-the-play, Hamlet's effort seems more directed to embarrassing both Gertrude and Ophelia with ranting and crude jokes than to work on getting Claudius nervous—but the play being acted before them is supposed to do the latter. One might ask if his focus is more to torment Gertrude (and Ophelia) and to express his general misogyny than to finish off Claudius.
After this, instead of dispatching the king, Hamlet first goes through a prolonged mind-games session with Rosencrantz and Guildenstern, then says to himself, as he sees the king in prayer, that he can't kill him because Claudius's soul will then fly straight to heaven. You might consider if Hamlet really believes this, or is using it as an excuse because he'd rather kill Claudius "in th' incestuous pleasure of his bed," where Gertrude would witness the murder (and Hamlet would then be able to witness what is known as "the primal scene," though Claudius is only his uncle and step-father rather than his real father).
In act 3, scene 4, Hamlet cries, "Mother, mother, mother!" like a child almost, in answer to the request to see her she's conveyed through others. The scene between them that ensues is enormously Freudian. Hamlet pours out a stream of abuse at her. One might ask, is she the focus of his hatred and revenge more than Claudius? Has Hamlet really taken leave of his senses, or is it still part of his elaborate ruse to make them think he's crazy?
In the final scene when the catastrophe occurs, Gertrude upon being poisoned calls to her son in her last breath. Hamlet is enraged by the poisoning of her and, being told the rapier's point is "envenomed too," finally finishes off Claudius. In this last moment, have Hamlet and Gertrude finally been reconciled? Is the normal relation of mother and son restored by the treachery of Laertes and the King?