In order to level your character up, you need "experience".

Experience can be gained in numerous different ways:

• Killing creeps.
• Killing neutral creeps.
• Destroying towers.
• Killing heroes.
• Slaying bosses.

Experience is divided by the amount of heroes close to the unit when it dies, this is why when people solo lanes, they gain twice the experience that the other heroes do. As you level up, it requires more and more experience to level up.

Currently, (V6.0+) the level cap is 18 and this requires 17,000 experience to reach.