This achievement is pretty self explanatory, you must shoot him in the heart and you can't use Vigors to help you. The first time you encounter a Handyman is in the Plaza of Zeal, just after you leave the Gunsmith's shop.
However there is an old glitch from the 360 version that made it's way into Xbox One version. As you exit the shop, turn right and the Handyman will spawn in. If you look above there's a Skyhook you can attach to. Jump to that and then jump to the roof on your left. The Handyman will jump up to you, quickly jump back over to the Skyhook and dismount and if done properly the Handyman will be stuck up on the roof. You now have a clear shot at his heart.
It can take multiple tries to get him stuck up there but keep at it. I've included a video I found on YouTube that shows you where to also get a Sniper Rifle if you want to be 100% sure that you're hitting his heart, but I did it with the Hand Cannon without any problems.