Act 3 Scene 3

After Hamlets bizarre behavior during the play, Claudius orders Rosencrantz and Guildenstern to take Hamlet away to England. Meanwhile, Polonius offers to spy on Hamlet as he talks to Gertrude. Hamlet then later catches Claudius by himself, but restrains from killing him because he's in prayer, which would send him directly to Heaven. He descides to hold off until Claudius is in the middle of a sinful act, as to send him straight to hell.

At this stage, I believe Claudius is very fearful for his life. He's afraid of Hamlet now, and sees him as a threat. "I like him not, nor stands it safe with us to let his madness range." If I were in his shoes, I'd probably bussle down to the church too and try to repent my sins. I believe he's still too flushtered to be able to concentrate on his praying, which is why after Hamlet leaves he says:
"My words fly up, my thoughts remain below;
Words without thoughts never to heaven go."


During this scene, Hamlet passes up a golden opportunity by sparing Claudius's life.

"No. Up sword, and know thou a more horrid hench."

If Hamlet had killed Claudius right then and there, he wouldn't have had reason to kill anyone else. Many lives could have been spared if he'd killed the source right then and there.