When we talk about the play's "tone," we're talking about mood, atmosphere, and/or the play's attitude toward its subject matter. For the most part, Hamlet happens to be a very dark and introspective play that's full of anxiety and uncertainty. This is mostly because Hamlet, our protagonist, is dark and introspective and full of anxiety and uncertainty. When Hamlet delivers all those introspective and tortured soliloquies about mortality, betrayal, and the futility of life, he creates a somber atmosphere that allows us to see the world from his point of view.
Hamlet's tortured mind isn't the only thing that creates the play's dark mood. If we think about it, Shakespeare sets the tone in the play's very first line – "Who's there!" That's what Bernardo says one dark night on the castle battlements. The guards are on edge, a bit fearful, and full of anxiety. Why's that? Because a ghost that looks a whole lot like the dead king has been paying them a visit each night. Pretty spooky, wouldn't you say?