The thing to remember in answering this question is that Hamlet takes no immediate action simply because of the recent passing of his father. He responds to his father's death, and his mother's controversial marriage to his uncle, with sadness and anger, but he takes no action. His soliloquy in Act I, Scene 2 provides some insight into his mood:
How weary, stale, flat and unprofitable
Seem to me all the uses of this world!
Fie on't! ah, fie! 'tis an unweeded garden
That grows to seed; things rank and gross in nature
Possess it merely. That it should come to this!
Hamlet first begins to plot against Claudius when he encounters his father's ghost, who tells him that his uncle murdered him. Even then, however, Hamlet does not act until events have left him willing to risk all. The deaths of Polonius and Ophelia, Claudius's plot to have him murdered by the English, and the challenge made by Laertes seem spur him into action, though he still has no real plan to avenge his father's murder, only attacking him after he realizes he has been poisoned. The point is that Hamlet is not really notable for his actions in response to the recent passing of his father, but rather his inaction. He is sidetracked by frequent philosophical reflections on the evil in the world, and, at least until he stages the play, lingering doubt as to whether his uncle was indeed guilty of murder. So his plot to uncover his uncle's guilt and avenge his father seems, in the context of the play, to be entirely justified.