Your hero has limitations when upgrading level. While your captains can attack any squad on the Kingdom map and explore crypts, your hero can only attack Rare and Heroic squads, or enemy players.
Hence the fastest way to level up your hero is to attack Rare monster squads. You will later be able to attack Heroic squads with all of your captains and hero once you gain sufficient might and level. You might want to use the Watchtower to quickly find the Rare squad you wish to attack.
You can also gain experience by opening experience chests that may be purchased in limited offers.
Experience from completing achievements from "Achievements" is awarded to the Hero, regardless of where he is at the moment, and whether he is alive or dead.