One of the more famous lines in Hamlet is, "To thine own self be true, and it must follow, as the night the day, thou canst not then be false to any man" (1.3.84-86). Which of the characters in Hamlet are true to themselves? Does that prevent them from being false to one another? Does the meaning of this quote change for you when you consider that it's spoken by Polonius, one of the play's most frequently mocked characters?

The steady and loyal Horatio may be the only character in the play completely true to himself, but I would argue that Hamlet and Laertes are also largely true to themselves. Hamlet's hesitation to kill Claudius arises from following his own heart: it is not in his nature simply to rush out and seek revenge until he has verified the facts of the story for himself. He doesn't want to kill an innocent man. And true to his nature, Hamlet does find a way to confirm the Ghost's story. Laertes, in contrast, is a person whose nature is one that is going to act rather than contemplate, which he does. The only place where he is not true to his own sense of honor—so yes, he is false to Hamlet, just as Hamlet is false in pretending at times to be mad—is when he accepts Claudius's underhanded ploy of fighting Hamlet with the poisoned sword. However, Horatio regrets it and accepts, when he is stabbed with the poisoned tip, the poetic justice of dying due to his own deception. Laertes states that he is

“justly kill’d with my own treachery.”

He then, true to his own nature, reveals all of Claudius's duplicity.
Given what a clueless, scheming balloon of hot air Polonius is, his words about being true to oneself are often read ironically. I agree that it is hard to take Polonius seriously when he says this. The words themselves may be true but are astonishing coming as advice from Polonius.