The crisis of a drama usually proceeds and leads to the climax. In Shakespeare's Hamlet, the proof that Claudius is guilty leads to Hamlet's decision to not kill Claudius while he's at prayer--and that is the climax of the play.
Hamlet, until he sees Claudius's reaction to the "play within the play," isn't entirely sure Claudius is guilty. He has no real proof--only the word of a ghost, who, he says in Act 2.2.565-572, could be a devil trying to deceive him (as, by the way, the witches do to Macbeth in his play of the same name). Hamlet needs proof. He is too reasonable to act like Fortinbras or Laertes and just jump into revenge without thinking it through.
The king's reaction to the murder scene in the play gives Hamlet the proof he needs, though, and he sets off to kill the king. He gets an opportunity but decides not to take it. Why? Because he thinks Claudius is confessing (he isn't, but Hamlet doesn't know that), and killing him immediately after he confesses his sins would send Claudius straight to heaven. And Hamlet doesn't want to send Claudius to heaven, not when his father is suffering in a purgatory-like state, and when Hamlet might be sent to hell because he kills Claudius.
The problem is, though, that when Hamlet decides not to kill Claudius because he doesn't want to contribute to his salvation, he is playing God. Salvation is God's business, not Hamlet's. Hamlet is messing where he shouldn't be messing.
The result--you see it in Act 5: the sight Fortinbras says doesn't belong in a castle, only on a battlefield. Death everywhere.
When Hamlet walks away from his rightful revenge, by playing God, he dooms himself and so many others. This is the climax. His receiving proof of Claudius's death could be considered the crisis, and Hamlet's refusal to kill Claudius while the king's at prayer is the climax. One leads to the other.