It might make a good beginning to identify the source of the quoted statement and to learn a little more about that source. The quoted text does not come from the play itself. The statement isn't "from Hamlet" or even from Shakespeare's pen; it's from the related field of Shakespeare criticism.
In the four centuries since Shakespeare lived, wrote, and produced his seminal plays and poems, many scholars have devoted a substantial portion of their careers to analyzing Shakespeare's works within the contexts of philosophy, Elizabethan society, and world literature.
Perhaps the best-known Shakespeare scholar of the early 19th century is the British author, essayist, humanist, literary critic, art critic, philosopher, painter, and social commentator known as William Hazlitt. The quoted statement you've been asked to discuss sounds like Hazlitt, but let's be sure. I would suggest verifying or disproving this assumption, and then learning nearly as much about the source of the quoted observation as you learn from direct study of the play itself. At least learn as much about the critic as Wikipedia will yield.
Then read or re-read Hamlet not with the singular goal of proving the merits of this critic's observation, but rather with an equal mind toward disproving or falsifying the claim. Could these various features of Hamlet's personality just as easily imply some other proclivity besides a striving for moral beauty? Is there another way to read or interpret the titular character?
It's fine if you agree entirely with Hazlitt (or whatever critic made the quoted observation) but make a good case for it. When you discuss the merits of the statement, be likewise willing to admit any peculiar flaws.
Could you find evidence that Hamlet is motivated by something other than a striving toward moral beauty?
Could his revenge motive, for example, be more base and personal than the term "moral beauty" would imply? How good is Hamlet, after all? What is it that drives him? Not just according to Hazlitt or another esteemed critic, but according to your own estimation. Is it possible that the critic who first spoke of Hamlet's moral beauty is more idealistic about human nature than Shakespeare? Does Shakespeare's text in all ways fall into accord with the critic's observations? What do you think?
When you discuss the merits of a critical argument, it is important not to be uncharitable, of course. According to the principle of charity as it applies to philosophical argument, you want to be very careful that you do not misconstrue the argument you're criticizing by rendering the weakest or most exaggerated possible reading or interpretation of that argument in order to poke easy holes in it (The Strawman Fallacy). But likewise, you don't want to take a critical perspective from another as sacred, inarguable, given, just because it emerges from an esteemed source or the distant past or because sounds like an obvious truism (Argument from Authority).
In short, be sure to discuss both the merits and the flaws of reading the source material through the lens of this particular interpretation. And -- to paraphrase another respondent to this question -- be sure to justify your various observations by citing specific examples form the source text. Don't lean too much on generalization.
Could you find passages in Hamlet where the implication seems clear that his overall imperative is a striving toward moral beauty? Could you find any passages that might yield a contrary insight into the protagonist's motivations? When you weigh the moral implications of all of Hamlet's thoughts and actions for yourself, do you finally agree with the earlier critic, or not?
Thinking it through for yourself and then demonstrating your reasoning in clear prose, showing the reader how you've weighed your own interpretation based upon all the evidence at hand: these are the essential actions.
