How does Hamlet figure out that Claudius killed his father?

In act 1, scene 5, Hamlet is able to speak with the ghost whom his friends have seen wandering at night. The ghost identifies himself as Hamlet's father and says,

Now, Hamlet, hear.‘Tis given out that, sleeping in my orchard,A serpent stung me. So the whole ear of DenmarkIs by a forgèd process of my deathRankly abused. But know, thou noble youth,The serpent that did sting thy father’s lifeNow wears his crown.

If the murdering "snake" now wears his crown, the murderer must have been Claudius. The ghost then tells Hamlet that if he loved him at all in life, he should avenge his death. However, Hamlet is not a murderous being by nature and doesn't know whether to believe that this ghost is really his father or that it is telling the truth about Claudius. He monitors Claudius closely for a couple of acts, trying to throw him off with feigned insanity so that his real purpose will not be discovered. Finally, he decides to work with a group of actors and have them devise a play very closely aligning with the murder Claudius allegedly committed. He plans to watch his uncle's reaction to the play and then determine his course of action.
As the actors on stage reenact the story the ghost told Hamlet in act 1, the following transpires:




OPHELIA
The king rises.




HAMLET






What, frighted with false fire?




GERTRUDE






How fares my lord?




POLONIUS






Give o’er the play.




CLAUDIUS






Give me some light, away!







Claudius immediately leaves in anger and doesn't return. It is at this moment that Hamlet is certain that the ghost has told him the truth and that he must avenge his father's death.

In act I of Hamlet, the ghost of Hamlet's deceased father, King Hamlet, tells Hamlet that Claudius killed him:

Upon my secure hour thy uncle stole, With juice of cursed hebona in a vial, And in the porches of my ears did pour The leperous distilment. (I.v.61–64)

The King's ghost announces that Claudius, Hamlet's uncle, poured poison in his ear while he was sleeping. The Ghost also implores Hamlet to seek revenge for his foul murder. Even though the Ghost seeks revenge, Hamlet is still not quite certain if the Ghost's words are true. Later in the play, Hamlet has the players re-enact the murder scene, and when Claudius is upset and has to leave the room, Hamlet feels he has proof of the Ghost's words and Claudius's guilt.
However, as the play progresses, Hamlet does everything but seek revenge. He feigns madness, which causes even more problems, such as the death of Polonius and, indirectly, Ophelia's demise. By the end of the play, it is clear that Claudius is guilty of the murder when he plots to kill Hamlet with a poison-tipped sword when he duels with Laertes. (If that fails, he plans to have Hamlet drink poison from a cup.) Ironically, things don't work exactly according to plan, and the scene ends with the deaths of Claudius, Gertrude, Laertes, and Hamlet.

Initially, Hamlet is told by the ghost of his father at the beginning of the play that Claudius murdered him in his sleep. King Hamlet's ghost tells Hamlet that while he was sleeping in the apple orchard, Claudius poured poison into his ear, which ended up killing him. Hamlet is shocked and outraged to hear that Claudius is responsible for his father's death and vows to get revenge on his uncle. However, Hamlet hesitates and begins to question whether the ghost he saw was actually his father or a demonic spirit.
In order to ensure that King Claudius killed his father, Hamlet conspires to have actors reenact the circumstances of his father's murder during their performance of The Murder of Gonzago. In act 3, scene 2, Hamlet tells Horatio to watch Claudius's reaction closely to witnessing "The Mousetrap" scene. Sure enough, King Claudius is visibly upset and immediately leaves the play after witnessing the murder scene. King Claudius's reaction is telling and is the proof that Hamlet needs to ensure that his uncle murdered his father.