Hamlet is surely one of the most complex and well analyzed characters in literature. To describe him in two or three words is challenging, but consider these. He is conflicted, introspective, and melancholy.
Hamlet is a very conflicted character. He loves Ophelia, but must push her away from him--for her welfare as well as his own, as he tries to determine the truth of his father's death. In regard to avenging Old Hamlet's death, Hamlet is conflicted between performing his duty as his father's son (exacting a revenge killing) and actually committing the violent act.
Hamlet is intensely introspective. He continually examines his own life, his thoughts, his actions, and his motivations for those actions. He tries to rationalize his actions or inaction, but he also condemns them, and himself.
Hamlet is indeed the melancholy Dane. Sadness permeates almost every thought. He is surrounded by death and obsessed with the idea of death. His famous speech in the graveyard as he holds the skull exemplifies this.