In what respects is the 'Play Scene' (III.ii) a turning point? How is this turn compounded by Hamlet's actions in the 'Closet Scene (III.iv)?
Act III is ALWAYS the climax of Shakespearean tragedy. The play, as my collegue has stated, gives Hamlet the proof he has been needing in order to act out his vengence against the king. Up to this point, Hamlet has doubted the truthfulness of the ghost, whom he thinks might be the devil trying to damn him by tricking him into murdering Claudius. Now he knows what action he must take.
In his mother's closet, Hamlet mistakingly kills Polonius, thinking he is the king. Instead of exacting his revenge, he complicates the situation. Now he has committed the same sin as Claudius, though not intentionally. This rash action leads to the deaths of a number of characters--including Hamlet, since now Leartes is justified in wanting his own revenge.
Had Hamlet not been so moved by the ghost's accusation and Claudius' reaction to the play, he may have acted out his revenge more carefully. It is ironic that Hamlet, who heretofore has been hesitant to act, now acts so suddenly and with such dire consequences.