Story Mode is a fairly lengthy campaign, featuring all the games characters. You'll play a series of matches, with many cinematics telling the story of Injustice in between. All you have to do is play through and finish the game to unlock this achievement.
Difficulty doesn't matter, so you can change the Story difficulty to 'Very Easy' in Gameplay Settings from the main menu and you can pretty much mash through the game. If you ever lose a fight, you can hit 'retry' and start from where you left off. Your current progress can be checked by viewing Chapter Select or your Hero Card.
You will also unlock "Almost There" at the halfway point of Story Mode.