One internal conflict is his seeming fear of action; this results from his confusion over expectations of the Ghost and expectations of his religion. He wants to avenge his father's death, but questions whether or not the Ghost is truly his father. Because of this question and his religious beliefs, he suffers from inaction, from his failure to make a move. He eventually compares himself to Fortinbras who can make a decision and act almost instantly, and this causes him considerable distress.
He is also struggling internally with his love for his mother. On the one hand, he truly loves his mother; on the other, he despises her for marrying her husband's brother and so soon after his father's death. Her actions condemn her in the eyes of their Church and doom her in Hamlet's eyes
He is also having trouble dealing with Ophelia's betrayal. Again, he loves her but is disgusted by her behavior: her trickery in refusing to keep his tokens of affection and her unwillingness to be honest with him reveal her betrayal. For all of this, he reacts violently and cruelly, only to find out later that he has made a grave mistake.
In all, Hamlet is suffering internally. His father, mother, uncle, and lover have all caused him to doubt his own decision making, and for this he is thrown into terrible fits of internal depression.
Hamlet appears to struggle with a sense of self-doubt. This is seen in his struggle with acting on his plot of revenge. Hamlet has vowed to avenge his father's murder, but for much of the play he does not act on it. This causes him to struggle with himself. At times, he curses his inability to act. Hamlet is more of a philosopher than an action oriented man, he tends to think things through and wonder at what motivates action than actually going through with something.
Hamlet seems to struggle with depression, as well. His thoughts on the people in his life, and life itself, are often dark and negative(although-rightly so at times!).