A user on Reddit was able to find out the names and descriptions for every Dark Ops Challenge in Black Ops 3. From this we learned that Dark Ops Challenges can be earned not only in Multiplayer, but also in Zombies.
1) Obtained – Earn Dark Matter by earning Diamond Camo on all weapons in the game.
2) 100 Percenter – Complete 100% of all multiplayer challenges.
3) Trick Shot – Destroy an HC-XD with a Combat Axe.
4) Dr. Lung – Kill an enemy underwater when you are both close to drowning.
5) Relentless Killer – Earn 1 Relentless medals (20 kills without dying).
6) Brutal Killer – Earn a Brutal medal (25 kills without dying).
7) Nuclear Killer – Earn a Nuclear medal (30 kills without dying).
8) Nuked Out – Earn a Nuclear medal in FFA without using scorestreak rewards.
9) Finish the Job – Kill an enemy that survived damage from 5 teammates in the same life.
10) Calamity – Kill an entire enemy team of 6 or more players in a very short time with a Specialist weapon.
11) Frenzy Killer – Earn a Frenzy Kill medal (5 rapid kills).
12) Super Killer – Earn a Super Kill medal (6 rapid kills).
13) Mega Killer – Earn a Mega Kill medal (7 rapid kills).
14) Ultra Killer – Earn a Ultra Kill medal (8 rapid kills).
15) Chain Killer – Earn a Kill Chain medal (9 or more rapid kills).
16) Fresh Start – Perform a “Fresh Start” stats reset.
1) Deresurrectionist – Reach Deressurectionist (level 35).
2) Apocalypse Averted – In Shadows of Evil, help the others.