Did Hamlet actually mourn the death of his father, or just the idea of his father?

Certainly Hamlet idealizes (and really idolizes) his father. Multiple times in the play, he compares his father to Claudius, most memorably in his first soliloquy, when he says that comparing King Hamlet to his usurper brother is like comparing "Hyperion to a satyr." He also bemoans the fact that his mother "my uncle, My father's brother, but no more like my father/Than I to Hercules." So clearly Hamlet admires his father. Though his procrastination in avenging King Hamlet's death is one of the play's central plot features, we should note that he swears to do so out of a strong sense of obligation to his father. Whether Hamlet worships an ideal of his father is a difficult question to answer. It is perhaps true that many young people idealize their parents, especially if they lose them early in their lives. What is clear is that when the audience is introduced to Hamlet, he is overcome with grief at his father's passing and his mother's decision to remarry. When he is charged by his father's ghost with avenging his untimely death, he is driven by love, admiration, and duty to do so. So his admiration for his father, and his grief at his passing, is clearly authentic and what one might expect from the son of a king.