For this, you need to complete a level using only running attacks. Although the game tells you to press the stick to run, it is an option (and much easier) to use the right trigger to run.
I can also confirm that you can accidently use a regular attack as long as you don't hit an enemy with it. At one point, I accidently used a basic
attack. It didn't make contact with an enemy and the achievement still unlocked at the end of the stage.
I managed this on the Studio stage.