You can upgrade your heroes, and to do so, you first need to find a hero card. The first legendary (hero) card you find unlocks the hero for you to use and upgrade to up to level 33. The next card of the same hero lets you upgrade them up to level 66, and the 3rd card of the same hero enables you to upgrade it up to level 99. 
You can find these cards in daily, golden, and epic chests. Alternatively, if you want to ensure that you get a hero card, you can buy the new special hero chest available in the in-game shop after you reach rank 17 on your account. However, it's only available for purchase once.