In act 2, scene 2, Polonius reveals his belief that the cause of Hamlet's madness is that Hamlet is in love with Ophelia. Polonius previously told Ophelia (his daughter) to break things off with Hamlet, and Polonius now tells Claudius and Gertrude that this is likely what precipitated...
In act 2, scene 2, Polonius reveals his belief that the cause of Hamlet's madness is that Hamlet is in love with Ophelia. Polonius previously told Ophelia (his daughter) to break things off with Hamlet, and Polonius now tells Claudius and Gertrude that this is likely what precipitated Hamlet's mental illness. They plan to allow Ophelia to speak with Hamlet and observe their interaction, and Polonius says,
If he love her not
And be not from his reason fall'n thereon,
Let me be no assistant for a state
But keep a farm and carters (2.2.165–168).
In other words, Polonius stakes his profession on his certainty that Hamlet has lost his reason as a result of his unrequited love. Of course, we know that this is not true. Hamlet is not actually mad (at least according to Hamlet himself) but has resolved to act as though he is in order to investigate Claudius's involvement in King Hamlet's death. Therefore, the audience knows a great deal more, here, than Polonius and the royal couple.
Later, in act 3, scene 3, Hamlet happens upon Claudius while the king is praying. It occurs to Hamlet that he could murder the king now, but he decides not to because it wouldn't be fair. Claudius killed King Hamlet before he could confess to or atone for his sins, and so King Hamlet went to Purgatory; Hamlet doesn't want to send Claudius straight to heaven (since he has just, perhaps, sought absolution), because this is not true revenge. Hamlet doesn't realize, however, that Claudius is actually praying unsuccessfully—he does not actually feel enough remorse to give up what he gained by murdering the old king. In the end of the scene, he says to himself,
My words fly up, my thoughts remain below.
Words without thoughts never to heaven go (3.3.98–99).
Thus, we know more than Hamlet. Hamlet actually could have killed Claudius at this time, and Claudius would have met death with all his sins on his head.