Below is a list of all the achievements in the game and how to achieve them.
Storyline Achievements
Origin Missions
Alternative version of…
Reputation Achievements
Other Achievements

Storyline Achievements

Who’s Laughing Now?
Complete Batman vs Joker mission.

Men of Steel
Complete Superman vs. Lex Luthor mission.

In Brightest Day
Complete Green Lantern vs. Sinestro mission.

Assassin in Disguise
Complete Batman vs. Ra’s Al Ghul mission.

Watching the Watchtower
Complete Cyborg vs. Deathstroke mission.

Superior Speedster
Complete Flash vs. Zoom mission.

Only A Scarecrow
Complete Batgirl vs Scarecrow mission.

Krypton Feud
Complete Superman vs. General Zod mission.

Tsunami? Tsunam-you!
Complete Aquaman vs. Ocean Master mission.

Found You!
Complete Robin vs. Harley Quinn mission

Clash of the Titans
Complete Wonder Woman vs. Cheetah mission

Honorary Superhero
Complete the final mission.

Origin Missions

These achievements are pretty straight-forward. If you have 1500 reputation, you can buy a origin mission to Batman, Superman, Green Lantern, Flash, Wonder Woman and Aquaman.

Meet Bruce Wayne
Complete Batman’s origin mission.

Meet Kal-El
Complete Superman’s origin mission.

Meet Hal Jordan
Complete Green Lantern’s origin mission.

Meet Barry Allen
Complete Flash’s origin mission.

Meet Diana Prince
Complete Wonder Woman’s origin mission

Meet Arthur Curry
Complete Aquaman’s origin mission

Alternative version of…

Open your notebook and type in your hero (Batman, Wonder Woman etc.). Now, select any option except for the first one. You should get your achievement by doing this.

Not Your Normal Batman
Spawn an alternate version of Batman.

Not Your Normal Superman
Spawn an alternate version of Superman.

Not Your Normal Flash
Spawn an alternate version of Flash.

Not Your Normal Wonder Woman
Spawn an alternate version of Wonder Woman.

Not Your Normal Aquaman
Spawn an alternate version of Aquaman.

Not Your Normal Cyborg
Spawn an alternate version of Cyborg.

Not Your Normal Green Lantern
Spawn an alternate version of Green Lantern.

Not Your Normal Martian Manhunter
Spawn an alternate version of Martian Manhunter.

Reputation Achievements

These achievements requires 3000 rep of all three types.

Gotham Baller
Get 3000 amount of Gotham reputation.

Kind of a Big Deal
Get 3000 amount of Justice League reputation.

Super Popular
Get 3000 amount of Metropolis reputation.

Other Achievements

The More You Know…
Simply open the Batcomputer and select an object to spawn.

Forging Justice
First, go to the batcave. Now, press the bubble (scissors and glue) on Alfred’s head to enter Hero Creator. Make any hero and save it.

New Skin
Open the Batcomputer and select the tab with Maxwell’s head and a lock. Buy any costume.

Crafty
Create Gizmo and give him metal.

Sea Beast Boy
1. Go to Atlantis. 2. Create Beast Boy.

All In a Day’s Work
Create Alfred and Dirty Batmobile. Let Alfred do rest of the work.

Brains!
1. Create something like undead plague or just a zombie. Let them infect the whole city.
2. Seach for a location where the people can’t run away, like one floor in the house. Spawn a zombie. Then spawn many ‘Green Lantern Corps’ and let them get infected.

The Main Man
Create Lobo. Then, create something with Sepulcher Black as the adjective, for example Sepulcher Black rock.

Feat of Feats
Finish all story missions first and complete all other achievements. If you still don’t have it, run around the entire map and complete the simplest tasks. Note that the simpler the tasks the better, do not go for tasks that require multiple objects.

Walk-In Closet
Get a ton of reputation and purchase multiple costumes.

No Evil Shall Escape My Sight
Spawn multiple “Green Lantern Corps” and then “Sinestro” and “Collective”.