Arm yourself with a pistol (it's all you will need) and get a cop car.
You can dial 911 and then jack the police car when it arrives, but this then gets you a wanted level which you have to get rid of before you can do anything else.
It's far easier to use the cheat code to spawn the FBI buffalo, as this will mean you won't get a wanted level for getting in the car. The code is 2275550100 dialled into your phone.
Once you're in the car, press LB to access the police computer, then select 'view current crimes'. Make sure to select a 'criminal on foot' crime, as with this you will only be up against one crook. If there's no 'criminal on foot' crimes to attend, just drive a block or two and then check the computer and there should be one showing then.
These missions are timed, and the time limits can be quite harsh so you will need to drive fast but carefully. Once you reach the crime scene, the time limit no longer applies - all you need to do is take out the crook. Arm your pistol and a headshot will do it. You will usually get more ammo than you used, and some cash as well, from the dead criminal.
Do the above 20 times in total, and you will get the achievement.
NOTE:- The cheat code to spawn the FBI car does NOT affect getting the achievement, however, the cheat codes for getting weapons does, so be sure not to use that one.