What is the character development for Hamlet in Act IV?
In Act IV Hamlet is becoming, if anything, more erratic and unpredictable in his behavior. Not only has he yet to inflict revenge on Claudius, he's actually beating himself up over it. We see this in Scene 4 when Hamlet wonders how it is that Norwegian soldiers can risk their necks for a meaningless patch of land while he still can't bring himself to finish off Claudius. Ironically, it's other characters in the play such as Claudius and Laertes who show more resolve in planning to kill Hamlet. The young prince's thoughts may be bloody, but his actions—towards Claudius, at any rate—are anything but.
Hamlet's chronic indecision in relation to Claudius stands in marked contrast to the impetuous resolve he displays towards others: he kills Polonius as well as sending Rosencrantz and Guildenstern to their deaths and (possibly) driving poor Ophelia to commit suicide. Throughout Act IV Hamlet veers back and forth between sudden outbursts of strange behavior and moments of deep contemplation and brooding. He was always a complex character, to be sure, but by the end of Act IV he's even more of an enigma.
Hamlet becomes more bitterly philosophical concerning fate and man's place in the world through the events of Act IV. In the previous act, Hamlet had actually made an attempt to kill Claudius, but stopped because he found Claudius praying, or so hamlet thought.
Unfortunately, in a fit of shock and alarm, Hamlet then slays Polonius, whose identity Hamlet did not know since Polonius was hiding behind the arras in Gertrude's room.
After being shipped off to England by Claudius as punishment for the slaying and while escaping the attempt on his life by switching the letter being carried by Rosencrantz and Guildenstern, Hamlet is captured by pirates and returned to Denmark. He has become far more cynical and worldly wise for having caught Claudius, Rosencrantz and Guildenstern red-handed in betrayal and a plot to assassinate him.
In the end of the play we can notice that Hamlet's reactions are becoming more obsessed and bitter towards Claudius and his mother. He even starts to speak in a more bitterness tone. Hamlet grows madder when he decides to describe the sexual encounters of his mother more in details: “Let the bloat king tempt you again to bed, /Pinch wanton on your cheek, call you his mouse, /And let him, for a pair of reechy kisses, /Or padding in your neck with his damn’d fingers,” (4.1.182-185).