Even in the Scene 3 (Act III), Hamlet is finally ready to act, having obtained the proof that Claudius has really arranged for his father's murder. However, he finds Claudius at prayer and once again refrains from action because he doesn't want to send him straight to heaven (as everyone who is killed during prayer will go to heaven, according to a common belief). So, Hamlet resolves to wait and kill Claudius when he finds him drunk, or angry, or in bed with his mother. He doesn't want to reward him. What Hamlet doesn't know is that Claudius can't really focus on his prayer, so it actually was a perfect moment to kill him.
In Scene 4, Hamlet demonstrates his readiness to act by killing Polonius, mistaking him for Claudius. This tells us that Hamlet isn't really a futile intellectual, incapable of acting and frozen in his own thoughts. He had only sought a proof that his uncle really deserves to die. Once he has this proof, he starts with his vengeance.
