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.