The achievement description is slightly vague. When a Prinny explodes, it checks for another Prinny in the explosion radius and then chains the explosion. The goal is to get ten chains rather than ten Prinnies. This can be done on the first stage, 1-1 with minimal mana, though you will need six movement items such as Slippers.
In the Senate, create 8 Good-for-nothing Prinnies. Their stats don't matter. You'll want one Prinny each with 7, 6, and 5 move instead of the original 4.
Position your Prinnies and human unit like this, skipping the crossed out friendly, as they aren't required.
Pick up the final Prinny and throw them into the bottom right corner of the map to set off the chain and get the achievement.