This solution is for all Psychopath achievements and was pulled from the 360 achievement guide done by Teaje
(there is not difference between the 360 version and the Xbox One version).
You will easily kill 10 psychopaths if go out of your way to every mission (or do Saint achievement), you just need to remember to keep full film on you and take a picture of the psychopath right before killing them off. I also recommend obtaining the Megaman blaster first so you can 1-2 shot the psychopaths.
I did this on my 2nd playthrough missing Kent and the Butcher make sure to take a picture of the psychopath(s) before going anywhere (I believe you can take a picture of the dead bodies) to get the photo pictures.