What are the benefits of leveling up the heros

According to what most people that opened a lot of packs said, there's quite a low chance of getting a legendary.

On average, people get 1 legendary per 20-25 packs. Some are less fortunate than that however.

Source: S1, S2, S3

The fastest way to level is hands down by playing games against real opponents. This can be done in either play mode or Arena.

Arena may be best (if you're doing well for gold or don't mind spending the money) because you get to use cards you wouldn't have and the wins and losses still grant you experience. The arena cards you get to use would grant you a stronger deck for that class and would give you (a theoretically) high chance to win than with the basic decks you have. Please note that you will only have the option between 3 classes in arena and may not get the one you need to level (credit: ChrisHateZ)

You can still create custom decks even at low levels. Whatever common cards you may have earned can help in this cause so you can build stronger decks. In play mode, you can gain the experience in either casual or ranked. There is no experience difference gained there, just your ranking.

Also, games that last longer (in the same level) generally give more experience than quick ending games in my experience.

The only reasons to play against the expert AI is to test a deck you are building before putting it out into real play and to complete the quest that requires you to defeat all Expert AI opponents (which will reward you will gold).