A list of the fights in the story, so you know how far you are:
Chapter 1: Batman
vs. Deathstroke, Bane, Lex Luthor (minigame), Joker
Chapter 2: Green Lantern
vs. Raven, Cyborg, Sinestro (minigame), Yellow Lantern
Chapter 3: Aquaman
vs. Flash, Shazam, Aquaman, Ares
Chapter 4: Joker
vs. Batman, Harley Quinn, Hawkgirl, Nightwing (minigame)
Chapter 5: Green Arrow
vs. Solomon Grundy (minigame), Killer Frost, Wonder Woman, Black Adam
Chapter 6: Cyborg
vs. Deathstroke, Lex Luthor, Catwoman, Cyborg
Chapter 7: Deathstroke
vs. Shazam*, The Flash, Killer Frost, Wonder Woman
Chapter 8: Batman
vs. Catwoman, Nightwing, Green Arrow (minigame), Yellow Lantern
Chapter 9: Lex Luthor
vs. Harley Quinn, The Joker, Hawkgirl, Shazam
Chapter 10: The Flash
vs. Yellow Lantern, Solomon Grundy, Sinestro, Green Arrow
Chapter 11: Wonder Woman
vs. Bane (minigame), Ares, Raven, Wonder Woman
Batman
vs. Batman
Chapter 12: Superman
vs. Black Adam (minigame), Aquaman, Doomsday, Superman
The achievement unlocks after the final cutscene.
*Almost There (50%) achievement pops here