High Chaos /Lethal way:
1. Once entered the meeting room (2nd floor in the Overseer compound), interact with the 2 glasses of wine on the table and swap them. Hide and wait until Campell walks in. He will soon die after drinking the poisoned wine, and the achievement unlocks upon the screen showing him eliminated.
2. You can always try the noisy way of using your weapons or gadgets to kill him.
Low Chaos / Non-lethal way:
Obtain information in the Archive (2nd floor in the Overseer compound) about a branding procedure, which can discredit Campbell. Afterward, get a brand in the Interrogation Room (same floor as the Archive). Next, render the Overseer unconscious with a sleep dart or choking. Finally, bring him to the Interrogation Room, bind him to the interrogation chair & brand him. Achievement will unlock upon the screen showing target neutralized.