Burn cards are earned after matches and for completing various challenges in the game. As you work toward all the other achievements, this will likely come naturally. You can earn up to three cards per match without any challenge bonuses, so you'll get it while working toward your 50 wins in each type. You can only hold 30 cards at a time, so make sure you are using them or discarding them (LT+RT) as they become available to make room for the next unlocks.