This achievement is earned by completing your training under beerus, the god of destruction (who was the guy from last years awesome Battle of The Gods Movie).
Beerus is a little different then other masters and you'll see why below. Firstly to unlock beerus you must complete the game, then you must reach level 70 and train with goku (I'll link my guide to Goku's training below). When you finish Goku's training beerus will be avaliable to train with but his spawn is pretty rare so use PQ43 and after every mission go for a run around the whole tokitoki city to find him.
When you find him you will begin your training,
00 - Morning Exercise - Reduce Beerus' health to a certain amount (Reward: Headshot)
01 - Feeling "Godly" - Reduce Goku's health to a certain amount. (Reward: Nothing)
This battle is against Super Saiyan God Goku and you get no skill or z soul for beating it.
02 - Master's Master - Reduce Beerus' health to a certain amount. (Reward: God Of Destruction Beam)
This ones different again as you have to first fight Whis, beerus mentor so your objective is to beat Whis as quickly as possible then the fight with beerus starts. This quest has a lose condition which is (Whis' health lowered after 2 minutes) and since I never failed this I assume you must finish before 2 minute mark (someone correct me in the comments please).
03 - Mastery - Reduce Beerus' health to a certain amount. (Reward: Sphere of Destruction)
You will finally face beerus here, just you vs him. Beat this and you'll unlock the achievement! Good job :)
- Beerus' spawn is random and will only happen after beating Goku's training.
- Beerus can stagger when hit unlike the main story line.
- Beerus has the strongest super in the game so if you want to go for all your moves achievement, parallel quest achievements, ect. I would save this training for last as keeping beerus as your mentor and using his super will allow you to activate z-assist when using a super which makes it more powerful. In short? Moves broken.
There is a way to finish his training much faster:
1) Equip headshot (and Beam of Destruction when you unlock it!)
2) Select PQ43
3) Select your custom character & beerus twice.
4) don't fight frieza, cell or buu. Just go for the dragonballs & finish the quest.
5) repeat steps 2, 3 & 4.
This is a method that will rank all mentors up faster, equip their moves and take 2 of them into the mission and you'll unlock their final test within 20 minutes each.
edit: People have reported that you have to spar with goku after his training is complete to find beerus although I didn't have to, it's best to add this if you're having trouble :)