Why are Hamlet and Claudius in conflict?
You'll find the Ghost of King Hamlet's explanation in Act 1, Scene 5. Here, he tells his son exactly how Claudius murdered him, by pouring a vial of poison in his ear while he napped, and how Gertrude, his wife, was complicit in the treachery:
.. Sleeping within my orchard,
My custom always of the afternoon,
Upon my secure hour thy uncle stole,
With juice of cursed hebenon in a vial,
And in the porches of my ears did pour
The leperous distilment; ....
Thus was I, sleeping, by a brother's hand
Of life, of crown, of queen, at once dispatch'd:
Cut off even in the blossoms of my sin,
Unhousel'd, disappointed, unanel'd,
No reckoning made, but sent to my account
With all my imperfections on my head:
O, horrible! O, horrible! most horrible!
If thou hast nature in thee, bear it not;
Let not the royal bed of Denmark be
A couch for luxury and damned incest.
Adieu, adieu! Hamlet, remember me.
Another reason, therefore, for Hamlet's conflict is his murdered father's plea for revenge, to be remembered, and for the traitors to be held accountable for their crimes.
Hamlet learns from the Ghost of his father that his uncle Claudius has poisoned his father. Not only that, but there seems to be some indication that something underhanded has already been going on between Claudius and Gertrude, Hamlet's mother.
Hamlet metaphorically and ironically says that the funeral baked meats were the cold meats for the wedding, it happened so quickly after his father's death. Were it not for the fact that Hamlet suspects Claudius of killing his father, there would still be tension between them because Hamlet has just come home to find his mother newly married and his uncle replacing, in every possible way, his dead father.
So that's two reasons for Hamlet to hate Claudius so far. There are two more reasons as well. First, Claudius tries to get Hamlet's friends to spy on him. Second, Claudius also tries to have Hamlet killed. For Hamlet, the conflict is based in avenging the evil done to his father. For Claudius, the conflict is self-preservation and greed. With Hamlet out of the way, nothing would stand in the way of his keeping the crown and the Queen and the power.
Hamlet is in conflict with Claudius because Hamlet's mother married Claudius, Hamlet's uncle, just one month after Hamlet's father had died. In addition, the Ghost of Hamlet's father reveals to Hamlet that Claudius murdered him, so this causes obvious conflict between Claudius and Hamlet.