The instructions for this achievement are pretty straight forward: just don't jump in this level. Avoid the holes, so you don't accidentally fall. Also, be sure to use the normal tile path, or if you take the crumbling tile path, they will dislodge on you and you will fail. Either than those two points, the gameplay mostly depends on the character you are using.
The Runner and Lizard are good. Although they are not the greatest for gliding across the gaps, they are both great for maneuverability. With this, you can move side to side and pretty much just dodge all of the gaps.
The Skater is also good for this achievement. Although he is not great with maneuverability, his speed allows you to cross the holes as big as one tile easily. If you are going extra fast by building up speed in the previous levels, you can perhaps glide across larger gaps.
Although not suggested, the Gentleman is pretty good too. What you can do come pretty close to a power cell, then intentionally fall into a hole. When you activate your electromagnet an instant after falling, the game will not count a jump and you will be able to skip a portion of the level.