Up until the "play-within-a-play" that Hamlet has done in the hopes of catching Claudius's guilty conscience, Claudius has no idea that the ghost has come and told Hamlet what Claudius did, and Hamlet doesn't really have much to fear from Claudius. Claudius certainly wants Hamlet's lunacy to be under wraps because it reflects badly on his kingship, but Hamlet is not regarded as a direct threat.
After the play, Claudius realizes that Hamlet DOES know the whole truth, and that completely changes the threat level. Once Hamlet accidentally kills Polonius, the danger to Claudius seems even more imminent. He even comments that "so it would have been with us had we been there." He knows full well that Hamlet is ready to murder to avenge his father's death. What makes the situation complicated for Claudius is that A) Hamlet is well-loved by the people of Denmark and Claudius thinks they will forgive him of any of his crimes. B) Gertrude will be out of her mind with grief should anything bad happen to Hamlet. Claudius's marriage to Gertrude will not be pleasant situation if anything happens to Hamlet and it is perceived as Claudius's fault. Claudius must get rid of the threat of Hamlet while not having any direct blood on his hands. His first plan is send him to England so that his craziness isn't a negative reflection on Claudius, but Claudius amends that plan and sends a letter asking for Hamlet to be killed upon his arrival in England. When that plan fails and Hamlet returns home, plan C is to manipulate Laertes's hatred and want of vengeance. Hence, those two devise the plan of a fencing match with poisoned swords and poisoned wine. It is ultimately a plan that backfires miserably killing everyone involved.
From Hamlet's perspective, his life becomes more complicated after he decides NOT to kill Claudius while he appears to be praying and again after he accidentally kills Polonius. By allowing Claudius to live, he lets him live on to create the threats that follow. Killing Polonius gives Claudius room to seek punishment for that dangerous and malicious action. Hamlet has "tipped his hand" that he knows the whole truth and that he is more than ready to kill in order to exact his revenge for his father's death. He is perceived as dangerous, and now Hamlet must act with much care and caution. He steals the letter to England away from Ros and Guil and then changes it in order to save his life. He makes a deal with pirates to return to Denmark. He cautiously enters in the fateful sword fight in Act 5.
The climatic moments of Act 3, lead to the falling action and complications of Act 4, and to the final, deadly, resolutions of all of the complications in Act 5.