As you may or may not already be aware, Runner 2 has three separate difficulties: Rather Easy, Just Right, and Quite Hard. Unlike in Bit.Trip Runner, where the difficulties merely altered the speed / collection requirements, the various difficulty levels in Runner 2 adjust the level itself, by adding enemies.
The other half of explaining this involves a brief explanation of the Perfect System itself - Collect all the gold and mode+ on the level (regardless of checkpoints, loops, or alternate paths), and you will score a perfect! If you reach the bonus cannon after crossing the finish line, you have gotten a perfect on the level. (Going on to then score a bullseye earns you a Perfect+, which is also tracked, but irrelevant to this question).
So a Double Perfect is earned when you have earned a Perfect on two of a level's difficulties, and a Triple Perfect when you have perfect!'d all three. Perfect!s are marked on the world map's levels as you achieve them, appearing as a colored exclamation point - Yellow ! for Quite Easy, Blue! for Just Right, and a Purple! for Rather Hard.
This means, to get the achievement you quote, you would have to beat 10 levels (2 in each world) a minimum of 3 times each (one for each difficulty)