It is important to remember that the advice that Polonius gave to his son Laertes before he returns to Wittenburg to pursue his studies is actually something that can be used to apply to many different characters, and your question astutely applies this advice to the category of Hamlet. I suppose there are plenty of ways you could argue Hamlet as a character is untrue to himself, but mostly they focus on his prevarication and procrastination concerning his inability to act in revenging his father. A great example of this is Act IV scene 5, when Hamlet, being sent to England by his uncle, is rebuked by the example of soldiers who are willing to fight for a piece of land that they have so little connection with. How much more should he then, who has so much more complaint, be willing to fight for vengeance:
How stand I then,
That have a father kill'd, a mother stain'd,
Excitements of my reason and my blood,
And let all sleep? while, to my shame, I see
The imminent death of twenty thousand men,
That, for a fantasy and trick of fame,
Go to their graves like beds, fight for a plot
Whereon the numbers cannot try the cause,
Which is not tomb enough and continent
To hide the slain? O, from this time forth,
My thoughts be bloody, or be nothing worth!
Hamlet clearly takes himself to task in this speech, comparing himself unfavourably with the soldiers that he sees and trying to goad himself into action, declaring an end to all the hesitation and doubt that is plaguing his mind.