Through natural story progression and gathering collectibles, I only needed to kill one more Red Cap at the end of the game.
A good spot to grind this, if you'd rather knock it out early, is in level five of Year five: A Giant Virtuoso. After getting the floo powder in Umbridge's office, you progress toward the first portion of the forest. There are five red caps here and usually, run right up to you if you walk near them. If they don't run up to you, they tend to pop their heads out of the ground, making them easy to spot. Once you kill all five, simply return to Diagon Alley and repeat the process until you unlock the achievement!