You can upgrade your heroes and to do so, you first need to find a hero card. You can find these cards in daily, golden, or epic chests. Because these are the legendary cards, they are very rare to drop. You can check the drop chance right in the store by tapping on each chest. This is what the hero card looks like:
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 lets you upgrade it up to level 99.