Initially, Hamlet conflicts with Claudius and Gertrude as a result of their marriage, which came so soon after the death of his father. When Claudius calls Hamlet his "son," Hamlet says to himself, "A little more than kin and less than kind" (1.2.64, 1.2.65). Claudius is Hamlet's uncle, and now...
Initially, Hamlet conflicts with Claudius and Gertrude as a result of their marriage, which came so soon after the death of his father. When Claudius calls Hamlet his "son," Hamlet says to himself, "A little more than kin and less than kind" (1.2.64, 1.2.65). Claudius is Hamlet's uncle, and now his stepfather, and Hamlet is less than enthused about this new relationship. His line indicates that he feels there are too many ties of kinship between them now. Further, when his mother, Gertrude, presses him regarding the fact that Hamlet is still mourning his father's death, he takes issue with the fact that she is not, and he says that he may show all the signs of still grieving,
But [he has] that which passeth show,
These but the trappings and the suits of owe. (1.2.85-86)
In other words, he may appear to be in mourning, but, more importantly, he feels more grief than he could ever show on the outside. He seems to imply that Gertrude may have seemed to grieve on the outside when her husband died, but her hasty remarriage proves that she didn't really feel her grief too deeply.
Later, after Hamlet has met and spoken with his father's ghost, his conflict with Claudius changes: he now blames his uncle/stepfather for his father's murder. When the acting troupe arrives, he lands on a plan to have them act out a play similar to old Hamlet's death. He says, "The play's the thing / Wherein I'll catch the conscience of the king" (2.2.583–584). Hamlet thinks that Claudius will provide a sign of a guilty conscience when he sees this play, if he truly is guilty.
Hamlet's major internal conflict has to do with his lack of action since learning of his father's murder from the ghost. He marvels at the actor who can feign his grief and resolve so compellingly while Hamlet has done relatively little to avenge his own real-life father. He says,
A dull and muddy-mettled rascal peak
Like John-a-dreams, unpregnant of my cause,
And can say nothing—no, not for a king,
Upon whose property and most dear life
A damned defeat as made? Am I a coward? . . .
it cannot be
But I am pigeon-livered and lack gall
To make oppression bitter, or ere this
I should have fatted all the region kites
With this slave's offal. (2.2.542–557)
He feels that he is a coward because he's done nothing yet to address his father's murder. He thinks that all he's done is mope around without even making plans for revenge. He wants to be a good son and be brave, but he doesn't feel as though he's been very loyal to his father.