For this achievement you need to go to Single Player> Multiverse> Battle Simulator, and complete one of those modes with all characters. I prefer the easiest, and first sim, which only puts you up against 5 fighters.
After defeating Brainiac in each of these, you will need to not skip the character ending. After each one, your progress on the achievement will increase.
You will need to complete the story mode before you can do this achievement, as doing so unlocks Brainiac as a playable character.
Edit, 5/23/2017
A couple of notes from the comments:
1. This can be done with your AI loadouts, just like any other Mutliverse events. When selecting your character, press
to select your loadout, then choose one of your AI loadouts and you won't have to fight... If that's your thing.
2. You can skip the end credits, and the repeated scene of
defeat. This will not nullify the achievement, and cuts down on the time spent staring at the screen.
3. It appears that using premium skins, i.e. Power Girl, Reverse Flash, John Stuart GL or Vixen, does not affect unlocking this achievement. I completed Supergirl's Battle Simulator with Power Girl and there was no issue.