The short answer? God.
The position of High Priest in ancient Israel started with God choosing Moses to represent Him to his fellow Hebrews (as they were then known in Egypt).
Moses didn’t ever become High Priest, though, and the reason was probably this:
Exodus 4:10-13But Moses pleaded with the Lord, ‘Lord, I’m not very good with words. I never have been, and I’m not now, even though you have spoken to me. I get tongue-tied, and my words get tangled. … Lord, please! Send anyone else.
And so God, fed up with Moses’ reluctance, appointed Aaron to do the speaking:
Exodus 4:16 Aaron will be your spokesman to the people. He will be your mouthpiece, and you will stand in the place of God for him, telling him what to say.
A year of more later, when the Israelites had escaped Egypt for the wilderness, Aaron became the first official High Priest.
The role of High Priest was inherited from Aaron and his descendants. To be a High Priest, you had to be direct male descendant of Aaron without any disability or other defect (even a broken arm would disqualify someone). You probably had to be older than 25. The only way to stop being High Priest was to die or sin.
As well as the rules for Levites generally, there were even stricter rules for the High Priest:
- must never leave his hair uncombed
- never tear his clothing
- never go near dead bodies (not even those of his own parents)
- marry only a virgin from his own clan (the Kohathites)—no divorcees, recovered prostitutes, or widows allowed.
The reason was that the priests should not bring shame on God’s holy name by their conduct or any other means:
Leviticus 22:32-33 … for I will display my holiness among the people of Israel. I am the Lord who makes you holy. It was I who rescued you from the land of Egypt, that I might be your God. I am the Lord.