There are two main ways the hero class can be leveled:
Leveling with Souls - like normal troops, hero's classes can be raised to Level 20 by spending souls. This provides them with basic stat gains. Reaching Level 10 also gives your hero class a home kingdom.
Unlocking Perks / Weapons with battle victories - a hero class can unlock perks at 10, 25 and 50 wins (although only one can be used at a time. 250 wins unlocks the hero class's special weapon.