One of the more intriguing challenges in understanding the uprising led by Korach is putting a finger on exactly what Korach's main complaint was. What was he objecting to and what did he seek to gain? The words of Korach himself don't provide very much detail but one of Moshe’s response is more telling. (16:10) "HaShem has brought you and your bretheren, the sons of Levi, closer and now you also want kehunah." It appears Korach desired to be a kohein as well.
However, Rashi (16:1) provides his own assessment of Korach's raison d'être. He was jealous of the appointment of Elitzaphan ben Uziel as the nasi of the children of Kehas. Korach staked a claim based on his father, Yitzhar, being older than Uziel. This, of course, begs the question: which one was it?
Gur Aryeh offers a reconciliation of Rashi's words and what Moshe later says to Korach. It was really the appointment of Elitzaphan that was fueling Korach’s revolt. However, his confrontation with Moshe could not survive without a significant following. He needed to bring a large group behind him in order to be able to make a convincing argument. If his platform was nothing but objecting to the appointment of Elitzaphan, it would have been seen as a completely selfish endeavor. He could not possibly gather a following. So, he made the movement about “the people,” the Levite people, at least. His campaign to oppose Moshe therefore became about the kehunah so that he could build a following of men who – at least as far as they were aware – shared his goals.
Perhaps another understanding can be offered. When Korach's confrontation with Moshe came to be, it was, in fact, about the kehunah. Rashi is not coming to explain what the final showdown was about but rather, how it came to be. What was it that fueled Korach to begin with? It wasn't as big the kehunah. Korach was upset about the appointment of Elitzaphan which, itself, might not have created such a significant kerfuffle. But, as jealousy tends to do, Korach's feelings got the most of him and led him to object to Moshe’s leadership on a much grander scale.