Reaper's Hellfire Shotguns
are his main weapons with 8 ammo, dealing UP TO
140 damage per shot with a headshot.
We will need to kill 3 enemies without reloading and having the kills register as "solo kills". Obviously, the perfect way to get this involves 3 enemies all being killed at 100% although these scenarios are rare. A solo kill is a kill that you did the most damage to (over 70), got the final blow on (name in kill feed), and did not get buffed by a teammate in any way (no teammates got assists).
Keep in mind that this achievements seems to be very inconsistent with many people reporting that they got 2 shotgun kills and then a death blossom kill (without reloading) to unlock it.
I recommend going for this while attacking and trying to kill enemies that newly spawned, or enemies sitting at the back that have low total healths. I personally got it by getting 3 "100" kills on Pharah, Junkrat & Hanzo inside the enemy base.
One good strategy to have is trying to get your first kill on a new clip, meaning that you will have 7 ammo to get the remaining 2 kills. Damage and enemy to low health, reload and finish them off to start the chain.
If you'd like to learn more about Reaper, you can view this video. The achievement is described and shown at 10:06: