This solution is now obsolete, as the achievement has been patched by the developers, to make it work as intended.
This solution is/was for the UNPATCHED, DISC VERSION of the game ONLY.
I'm putting this to just show what **I** did to unlock the achievement. I have the physical disc version.
I had to uninstall the game fully, and then install offline.
You have to go through the entirity of a mission, not just one chapter, with 5 or less resurrections.
I recommend going through with all three of the following guns fully upgrades and mastered with no vampire scopes. Mosin Rifle, MP44, Webley Magnum.
Take your time, destroy (by destroy, I mean, turn into meaty chunks) ALL bodies that you don't kill with a headshot, and DON'T use grenades, mines, etc. They can cause body damage kills and cause resurrections.
Run past zombies you don't have to fight.
I recommend the entire first mission.
Furthermore, when you're sure you've done the requirement, load up the start of the mission after you've finished, so the game saves. Then go back to the main menu from the pause menu, then quit the game completely by closing it on the dash.
Go back online. Get the update and, dependant on your connection speed, the achievement should pop whilst your console downloads the update. That's what mine did.