To unlock this achievement, no player can pick up a backpack throughout the run. Avoid any backpacks on pedestals, and when you defeat Granny the achievement will unlock.
This means that you cannot pick up coins or powerups, and can't use the shop at all.
The trick is having a second controller plugged in. At the beginning of each level, players exit the worm one at a time. If player 2 remains entirely inside the worm (not peeking out), and player 1 dies, player 1 can press A to respawn again. Thus you effectively have unlimited lives to complete the game with no backpack.