Claudius tries to comfort Hamlet over the death of his father by saying he doesn't just see him as a family relation but like a son. Hamlet admits that Claudius is now more than a family relation but he does not like him or his actions.
‘A little more than Kin and less than kind!’ (Hamlet, 1:2)
Hamlet doesn’t like the fact that Claudius has married his mother only weeks after his father died. Claudius also tells Hamlet not to grieve for his father so much.
‘To persever/ In obstinate condolement is a course/ Of impious stubbornness. 'Tis unmanly grief’
Towards the end of Act 1, the ghost of Hamlet’s father appears to him and tells him that Claudius murdered him. Hamlet is shocked and angry. His disgust towards his uncle remains constant throughout the play and drives him to seek revenge by killing him in the final scene.
‘O my prophetic soul!/ My uncle?’ (Hamlet, 1:5)
When Hamlet meets the ghost of his father, the ghost declares that Claudius murdered him. It could be argued that Hamlet may be suspicious of believing a ghost but his desire for this to be true seems apparent when he says
‘O villain, villain, smiling, damned villain!”
The wheels are set in motion for Hamlet to avenge his Father’s murder.
Having observed an actor deliver a monologue with true emotion, Hamlet berates himself for not being able to express his hurt and desire for revenge. He’s merely thought about revenge but not acted upon it. He steels himself and arranges for the players to perform a play about murder that shall provoke a reaction from the king and so prove that the ghost of his father was telling the truth.
‘The Play’s the thing/ Wherein I’ll catch the conscience of the King’
When Claudius storms out of the performance of 'The Death of Gonzago', Hamlet is finally convinced that the ghost was correct. Claudius murdered his father. When the prince goes to find Claudius, he observes him praying and Hamlet refuses to murder him while he’s praying.
‘O, my offence is rank. It smells to heaven/ It hath the primal eldest curse upon’t/ A brother’s murder’ (Claudius, 3.3)
At the end of his fight with Laertes, Laertes tells Hamlet that Claudius had set up the duel so that the young prince would be murdered. Hamlet’s thirst for revenge is total and he stabs Claudius and forces him to drink poison.
‘Here, thou incestuous, murderous, damned Dane. Drink off this potion’ (Hamlet, 5.2)