There are two options when purchasing the Battle Pass. If you go to the Item Shop in a Hub map you will see a Battle Pass display where you may pick up a crystal or a crystal head. The crystal is the normal Battle Pass and will cost 1,000 Raider Coins. The crystal head is also the Battle Pass but a bundle which includes the Battle Pass plus an instant level up of 25 levels and costs 2,800 Raider Coins.
To purchase the Battle Pass you will need to grab either the crystal or crystal head and raise it above your head until the purchase bar fills up completely. If you have enough Raider Coins the amount will be deducted from your balance. If you don't have enough Raider Coins, the game will prompt you to make a purchase with the payment method associated with your device.