You can only get this achievement once you've progressed so far throughout the main story missions and Warchiefs are available to fight.
Firstly you need to interrogate Captains or Worms to identify if a Warchief has any weaknesses.
Depending on his weakness, you then have to introduce that weakness into the fight. Some are easier than others.
If he's afraid of fire, draw him near a fire and weaken him until you can grab him. Once you can, walk or throw him through the fire.
The easiest one is if they're afraid of the bugs, draw him under the nest and shoot it with your bow and arrow.
You can also draw in a Caragor, either by dropping bait or bringing one with you but you're reliant on it surviving long enough for you to spawn the Warchief and get it close enough to him.
Once you've chased him down and killed him, the achievement will pop.
EDIT: As stated in the comments, I'll make this crystal clear. YOU have to kill him, don't allow an NPC or a wild beast to kill him.