In Act III, Hamlet finally started his plan of avenging his father's murder. The play he wrote, which closely replicated Claudius' murdering of Hamlet S.r., worked perfectly in causing Claudius to reveal his guilt to Hamlet. Hamlet had a chance to kill Claudius but didn't fearing that he would go to Heaven if he was killed during prayer. Hamlet wanted him to be damned for all eternity. It ends with Hamlet killing Polonius and warning his mother of what he is about to do.
We see in this act that Hamlet is not the mad, indecisive guy, but the cold-hearted, cunning prince who knows what to do and how to do it.
I think Hamlet will take his time and wait for Claudius to slip up before he kills him, since he now has Claudius on the defensive and frightened.