What does Hamlet reveal about himself that in believing the Ghost, he suddenly decides the play can prove Claudius' guilt?
In Shakespeare's Hamlet, regarding Hamlet's comment in Act II, scene ii, "The play's the thing / Wherein I'll catch the conscience of the King." (lines 599-600)
When Old Hamlet first appears to his son in Shakespeare's Hamlet, the Prince of Denmark is not sure if the ghost is a "good ghost" or an evil spirit trying to lure him to his eternal damnation by killing a king.
Elizabethans believed it was a mortal sin to kill a king, for only God could ordain who would be king. If Claudius has, therefore, killed his brother Old King Hamlet, then it was a mortal sin. However, if the Ghost is an evil apparition, then Hamlet feels he needs more proof. Hamlet is accused of being indecisive: this is his tragic flaw. However, it is easy to be more sympathetic to his cause when we remember that this young man doesn't want to forfeit his soul.
The Ghost's biggest complaint is that his brother killed him with sins on his soul, without the benefit of time to confess his wrong-doings. Because of this, Old Hamlet must wander somewhere between heaven and hell (purgatory?). It is this reason only that causes Hamlet to pause when, after the King's guilty reaction to the play, Claudius seems to be praying—Hamlet will not kill him then, even with his proof, because he doesn't want Claudius to be able to go straight to heaven, having just confessed his sins. (Ironically, Claudius cannot pray in this scene, though he tries.)
The quote that ends Hamlet's "O, what a rogue and peasant slave am I" soliloquy in Act II, scene ii, turns his attention from his failure to act, to his conviction that now is the time to act. He has been watching the players. He compares his reactions to those of the actors. They play their parts, only reading lines but without real feeling. He sees that he has been this way until now: having the information that his father was murdered, but unable to say or do anything, making his feelings useless because without action, they are meaningless.
With this in mind, Hamlet decides to get the proof or information he needs by having the actors play out a specific scene. He has heard that sometimes, when the guilty party sees his crime acted out in a play, he will react with such passion as to give away his guilt. Hamlet will have the actors reenact the murder of Old Hamlet, and Claudius' part in it. Based on Claudius' reaction, Hamlet will finally have proof to support or refute the Ghost's claim. If Claudius acts guiltily, then Hamlet will know the Ghost has spoken the truth, and then Hamlet can proceed without worry of losing his immortal soul.
At the end of the soliloquy, Hamlet announces that the play will be the tool he uses to stab Claudius' conscience into exposing this new King's guilt:
The play's the thing
Wherein I’ll see the conscience of the king. (599-600)