What is the best and fastest way to upgrade heroes?

Levelling up heroes requires EXP which is earned by using the heroes (i.e. attacking with your combat heroes in the APC) or using EXP items that you earn as rewards. Apart from spending money on packages and subscription in the game there is no short cut, just make sure you are using the heroes as much as possible to perform their actions and complete daily challenges and clash of zones challenges and any other events that occur that offer EXP items as rewards.