We see in Chazal different elements to the dispute of Korach.
- The pshat of the text seems to indicate that he used populist speech, we are all holy (i.e. taking away from the gedulah of Moshe and Aharon), but only as a pretext to become the head himself.
- The Talmud Bavli Sanhedrin 110A states that Yosef hid in Mitzraim 3 treasures, and 1 was revealed to Korach. In other words, he got his power from money that was not rightfully his. See also Targum Yonatan in Mikraot Gedolot for a slightly different version.
- The Zohar on Parshat Korach states that the Levi'im are from the Left, and the Kohanim were from the Right, and he wanted to take away the Kehunah, perverting the natural order of the world.