One of the central themes to kehuna is the kohen’s designation to torah instruction as described in the torah and chazal. Although neither ever disqualify a non-kohen from torah-instruction, both torah and chazal shine a favorable light on the kohen actively engaging in torah-instruction.
As a prelude to the kohanim functioning as torah instructors it is worthy to note the Rambam’s description of how Yaakov Avinu separated his son Levi from his other sons and appointed him to instruct and teach the ways of service to HaShem, to his brothers. He also instructed his sons to perpetuate this status of Shevet Levi -the tribe of Levi. (Rambam, Avodah Zorah 1:3)
The kohanim are in fact members of shevet Levi, and there are a number of instances where the Torah implies the kohen functioning in the realm of torah instruction;