What would be a Marxist approach to Hamlet's act 3 scene 4?
Marxist criticism is not simply an attempt to examine the existence of class within a literary work, but to understand its "forms, styles and meanings as the product of a particular history," as prominent critic Terry Eagleton puts it. Marxist readings analyze ideologies and structures within a text, assessing how social factors may impact the behavior of characters.
In this scene from Hamlet, a Marxist reading might be that Hamlet's own convictions, based upon the morals of the society in which he lives, lead him to believe that he has the moral high ground over his mother, whom he feels to be a sinner. This in turn means that Hamlet feels free to overturn the usual social expectation that he should treat his mother with respect: by sleeping with her late husband's brother, Hamlet feels she has resigned any position of respect she might have held. The early exchange between Hamlet and his mother feels extremely childish, with Hamlet's retorts to her sing-song sardonic echoes of her chidings ("Mother, you have my father much offended."). Note that Hamlet does use the "you" form in addressing his mother, but here this seems to signify distance or coldness, rather than the respect it might usually imply.
Hamlet declares Polonius to have been equally condemned by his behavior, an inhabitant of an "enseamed bed." To him, Hamlet uses the informal "thee" pronoun insultingly: "I took thee for thy better."
Later in the scene, as Hamlet is explaining his reasoning, the ghost of his father appears, but Gertrude does not see him, declaring that Hamlet is "mad." A Marxist reading here might be, then, that Hamlet's moral outrage is so extreme that it has literally manifested itself as a ghost only he can see, supporting his argument.
In Act 3 Scene 4 Hamlet murders Polonius hiding behind a curtain, thinking it was Claudius. Then Hamlet attacks his mother verbally for her disloyalty to his father. While he is yelling at his mother, the Ghost appears and reminds him that it is not his job to punish his mother, only to avenge his father's death. Gertrude thinks that her son has lost his mind because she cannot see the ghost so she believes he is talking to himself. The Marxist approach or view of Hamlet's situation might be that his state of mind is a reflection of his position as the avenger to his father's death, a job he never wanted. Marxism holds a belief that people's consciousness of the conditions of their lives reflects these material conditions and relations. Hamlet feels alienated from his family because his mother betrayed his dead father, murdered at the hands of her new husband. In other words, he is aware of his condition in life and so it is a reflection of his relationship now with his mother and his own self.