During the Plaza of Zeal, after leaving the Gunsmith for the first time.
There is a glitch that will make this achievement quite easy. You'll want to grab the sniper rifle from the back shop. Then use the skyhook on the left side of the street, where the handyman jumps down, and get onto the balcony. Have him follow you up there, and once he's up there, you want to jump down. He will get glitched there, and basically not move. Take your time to snipe him in the chest until he dies. If it doesnt work, just restart the checkpoint and try it again. Achievement unlocked!
Alternatively, you can get this achievement the way it was meant to be attained. Use the Sniper Rifle, located inside the shop at the back of the street. Now, let the Handyman charge you, and then he will either start a animation where he coughs or fixes his arm. During this animation, you'll want to scope in and shoot his heart before he is able to move. If you fail, restart the checkpoint. Also, sniper rifle upgrades really help make this easer, so consider getting them if you don't want to use the above method.