Psyclops are the floating black-and-blue like enemies that are typically found in Invaded levels. To earn this achievement, you must destroy twenty of them. For these kills to count, the level on which you eliminate one must be completed. Using the golden glove is very helpful in killing these enemies as if you are unable to bounce onto them correctly, they will activate a spike defense in which they release spikes all around their body, harming you if you are in the way.
The best level to grind these twenty kills on would be Level 4 invaded, titled Cake Walk (Invaded levels are levels that unlock after attaining 100 Lums in the original, easier level) as there are between 10 to 20 Psyclops that appear on the level in just one run. I was able to kill four of these on average per run, so it should only take 10 to 20 minutes to unlock this achievement, even if you are having trouble.
Here is an image of the Psyclops for visual reference: