Note that all 20 kills must be in the same ranked match for this achievement to unlock. Each match is 10 minutes long, so you do not have a lot of time to spare.
When you spawn, move forward so you can see and then hold down to select the closest minion spawn point and press until you have all of the minions. Keep holding down and use to select all of your red, green, and brown minions times so that you have 20 minions. Note that you may have to move slightly to be able to select all three minion spawn points. The blue minions won't attack an overlord, so you can safely ignore them. They won't hurt anything if you select them accidentally, however. When you have all your minions, aim at the base of your spawn point (large blue light) and press to place a rally point for your minions. Then cross the map to your opponent's spawn point. It is down the hill and up the other side of the little valley, and you should be able to see the glowing blue light from almost anywhere on the map. You will want to stand at or near the spawn point for the rest of the match to prevent your opponent from respawning here.
In the meantime, your opponent needs to come across the map to your original spawn point where your minions are waiting. Once you are at his starting point, have him walk up to your minions who will kill him in a matter of seconds. He will respawn right next to them, and merely needs to walk out and wait for his spawn protection to wear off. The invulnerability lasts fifteen seconds, and it takes about five seconds for them to kill an overlord, so you will get approximately 3 kills every minute once you have everything set up.
The achievement will unlock in-game when the 20th kill is reached.