Power-ups are purchased and upgrade from within the Store. There are three power-ups total:
Ghost Mode - Makes your character ghost-like and essentially invincible as you can go through any obstacle.
Ghost Sense - Slows down time and your character both, potentially giving you more time to react.
Undead Rush - Speeds up your character and makes him invincible during this time. You move rapidly and it is hard to collect or shoot anything while this power-up is active.
Each power-up can be purchased (brings it to Level 1) and upgraded to Level 5, with each upgrade costing progressively more coins/diamonds. Unlock all the power-ups first, which costs around 7,500 coins total (unlocks "Pack to the Hilt" achievement). Unfortunately, you will need over 1 million coins or more to upgrade all three power-ups to Level 5. See the "Grinding Method" at the beginning of the guide for more information on grinding coins. This is the most "expensive" achievement in the game.