In mission 8, you will be tasked with neutralizing Duke Luca Abele, Delilah's most powerful supporter.
To kill the duke, you only need to locate and positively identify him. Both the duke's location and that of his double are set dynamically. On different playthroughs, he has been found in the duke's chambers at the top of the palace, in the back garden, in his office on the third floor and in the throne room on the first floor (contains a Bone Charm)*. Two hints to identify the real duke:
1) Meagan will give you a hint about the behavior of the double. In each of my playthroughs, the double has been a smoker. Be careful, however, as this might not always be true.
2) Use the heart to listen to his secrets. The heart will give you enough information to identify the real duke. This will be easier if you've been paying attention to the lore as you work your way through the missions.
To incapacitate the duke without killing him, you'll be replacing him with his double. Steps to do this:
1) Identify and speak to the double. (See above.) The double will agree to replace the duke, but he needs the duke's medallion to prove his "authenticity" to the guards.
2) If the Duke is in the back garden, there is a hound nearby, so make sure the duke is far enough from the hound or that the hound has already been incapacitated. Otherwise the hound will alert.
3) Knock out the duke. If you're playing as Emily, Domino can be useful here as there are guards nearby.
4) Carry the real duke to the double, who will wait in the bedroom at the top of the palace. The rooftops are very useful for this, since no one pays any attention to what's above them.
Once you drop the duke in the appropriate location, a cutscene will trigger. The achievement will pop once the cutscene ends.
*If you find the duke or the double in another location, please drop a comment and I'll amend the solution.
Update: Edited to reflect feedback from SkepticalMario and TehAuthor