Hamlet is contemplative: Hamlet's contemplative nature dogs him throughout the play. He likes to think more than he likes to act. In a more typical revenge drama, the hero would have sprung right into action and attempted to kill Claudius. Not Hamlet. He wonders and weights his options. He wants confirmation that the ghost's words are true. And after he gets it, he still hesitates. As he says in act IV, he has been "thinking too precisely" about what to do. He tries to goad himself into action and states that too much thinking is three parts cowardice and only one part wisdom:
Of thinking too precisely on th' event—
A thought which, quartered, hath but one part wisdom
And ever three parts coward—I do not know
Why yet I live to say “This thing’s to do,”
Hamlet has a strong sense of justice: Hamlet is oppressed by the very idea that his uncle killed his father. It makes him want to kill himself. He is depressed by the rottenness and corruption he sees all around him in the Danish court. He says, "Something is rotten in the state of Denmark." He does not simply accept it as the way life is but is upset by it.
His sense of justice causes him to hesitate and fail to kill Claudius: even after the "play-within-a-play," when Hamlet gains reason to think that Claudius is guilty, Hamlet fails to kill Claudius because Claudius is at his prayers. Hamlet does not think it is appropriate that Claudius should go to heaven in a state of prayerful repentance when his father, killed while sleeping and therefore unrepentant, must wander the earth as a ghost.
Hamlet has a Christian worldview: Hamlet might be suicidal, but his fear of the Christian afterlife keeps him from killing himself. It also compels him to find out whether the ghost is telling the truth—the ghost may have been sent by Satan to tempt him into murdering an innocent man. As noted above, this fear keeps him from killing Claudius at his prayers. Hamlet fears sinning and then having to meet his Maker. He believes in sin and redemption.