The only way to increase Champion Levels are through fights, right? There's no Orb or anything to boost it faster?

As mentioned in the question, Class Trials are the alternative way to get champion XP. If you are very, very spendy, you can fully upgrade a class that way, but it is very expensive (5290 gems) and not recommended. So yes, you will have to make up the remaining champion XP via battles.

Class Trials for existing classes are scheduled every Thursday, for 24 hours. (New Class Trials operate on a different schedule, but there are no new classes anticipated any time soon.)

You can unlock the class quest from the Class Trial itself; you do not have to complete the corresponding kingdom quest first.