Lest we dismiss as preposterous Korach's assertion that the nation is “kulam kedoshim” and all should have an opportunity to serve as kohen, we need only remind ourselves of Hashem’s words, “V’atem tehiyu li mamleches kohanim v’goy kadosh,” charging us with to become a nation of kohanim and a holy people. The question is not what gave rise to Korach’s claim, but why indeed was he wrong.
Moshe’s challenged Korach, “U’bikashtem gam kehunah?” -- Do you, Korach, also dare desire serving as kohen as well? Given Korach’s mindset, it is hard, writes the Shem m’Shmuel, to understand why Moshe’s challenge would give him pause. Given his claim of supposed holiness, Korach might feel that in fact there is no one more deserving of the position of kohen than himself.
The Shem m'Shmuel explains that the position of kohen demands being “kadosh,” but not in the way Korach imagined. “Kadosh” means separate. The kohen was kadosh not because he separated from the people, but rather because he separated himself from his own base instincts and desires, including the desire for position and privilige. Aharon did not desire the role of kohen, but had it thrust upon him. He did not hold himself above and seperate from the people, but mingled among them, trying to mend broken relationships and trying to make peace where quarrels broke out.
Moshe’s argument to Korach was that the very notion of “u’bikashtem,” desire for the position, is antithetical to the role of a kohen. A person who is convinced of his own self righteousness to the extent that he sees only himself as deserving is someone who has not succeeded in separating himself from desire enough to deserve the role that he aspires to.
Rashi comments on “V’asu li mikdash” that the word “li” means “lismi,” for my [Hashem’s] sake. The difference between Hashem’s charge and Korach’s claim is that while Korach was motivated by his desire to serve, Hashem demands, “V’atem tehiyu li mamleches kohanim v’goy kadosh,” that our becoming a nation of kohanim and kedoshim be purely “li,” for his sake alone.