| Quote #1
From his very first scene, Hamlet sets himself up as someone who hates deception and values inner truth above all. Here, he insists that outward appearances (like his "inky" black clothing, sighs, and tears – all the common markers of grief) can't possibly "denote" what's truly inside him. In other words, Hamlet's saying that his anguish and grief over his father's death are far more intense than they appear to the outside world. He's also implying that Gertrude, Claudius, and the rest of the court are totally fake and disingenuous because they don't care about him or his feelings at all and are far too concerned with keeping up appearances.
| Quote #2
Polonius likes to dish advice, but he sure can't take it. Given Polonius's penchant for spying on his children and Hamlet in order to curry favor with King Claudius, he's not in any position to be talking about truth.
| Quote #3
Like Hamlet, the ghost dwells on Gertrude's "seeming" virtue. But is the ghost saying Gertrude cheated on him when they were married? Or, does the ghost merely see her remarriage as a betrayal? We get stuck on the meaning of "adulterate," which, in Elizabethan England could refer to a cheating spouse or any sexual sin in general (like incest). Either way, the ghost implies that Gertrude's remarriage retroactively makes their marriage into a sham.