So this sounds a lot harder than it is. "5 headshots," you might be thinking five sniper shots or something. Luckily, many types of "bullets" (could be Skye's little spiky projectile thingies or whatever) count and you can easily get this one with Viktor (who is unlocked from the beginning) and his assault rifle, or anyone who fires a lot of bullets in quick succession. I just used the example of Viktor because he's one of the starting characters. Just aim high and you don't have to kill them. You'll know you're getting headshots because of the "ding ding ding" sound effect and the little yellow cross-hairs that pop up.
As others have stated in the comments, other easy characters to get this with include Ruckus and Skye.
If you'd like a visual, here is a clip of when I got the achievement (you can see the in-game award for it pop in the top left):