Win all story mode minigames
Every minigame will automatically be played during story mode, but you must WIN the game in order to unlock the achievement. Every minigame is essentially hitting the correct buttons in enough time to prevent your opponent from harming you.
IF you fail to beat the minigame during your first playthrough of the storymode, DO NOT FEAR! Below are all the minigames, where they are located, and how you can know you beat them. Simply go to Single Player - Play Story - Chapters and then load up whatever chapter you didn't beat to unlock your achievement.
Alternatively, when you encounter a minigame during the story mode, if you fail, simply exit to the main menu, then choose to resume playing your story. You will be taken back to the beginning of the minigame to try again. - Thanks to The Heape for pointing this out.
You can also pause and unpause the game as many times as you like to see what buttons you need to press. It will not affect the achievement or minigame at all. - Thanks to The AuD Doc
Chapter 1 - Batman vs. Lex Luthor
Lex Luthor will drop down to one knee once you've defeated him
Chapter 2 - Green Lantern vs. Sinestro
Green Lantern will hit Sinestro with a bus
Chapter 4 - Joker vs. Nightwing
Joker will gain a full hand of cards and throw them all at Nightwing at once
Chapter 5 - Green Arrow vs. Solomon Grundy
Four rounds of shooting Grundy with arrows
Chapter 8 - Batman vs. Green Arrow
Completed once you've blocked Green Arrow's triple shot
Chapter 11 - Wonder Woman vs. Bane
Completed after the button mashing to block Bane's bullets
Chapter 12 - Superman vs. Black Adam
Simply watch the bar above the 3 Superman symbols. Doesn't matter if you get hit.
If there's something wrong with my solution, please message me or comment so that I may correct it.
Good luck and happy hunting!
Credit to PBreezey - TrueAchievements