Accomplish the indicated achievement to get the corresponding number of Gamerscore points:

Think About It... (20 points): Make first power destiny choice.
Destiny Begins (15 points): Select a Power.
Choose Wisely (20 points): Make second power destiny choice.
The Choice Is Made (20 points): Make third power destiny choice.
Got My Eye on You (30 points): Join Cyclops and the X-Men.
Magneto Is Right (30 points): Join Magneto and the Brotherhood.
Splicer (10 points): Equip your first X-Gene.
At Least It's Aerodynamic... (10 points): Equip your first suit.
I've Got the Power (20 points): Fully level up a power.
Fully Evolved (30 points): Fully level up all powers.
Four of a Kind (20 points): Equip a complete X-Gene set and suit.
Beginner's Luck (10 points): Complete your first Great combo.
This can't be happening! (20 points): Complete your first Insane combo.
How Strong Could It Be? (20 points): Trigger X-Mode.
Broken Glass, Everywhere... (20 points): Break 30 Combat Text Pop Ups.
Fight Terror with Terror (20 points): Defeat 10 enemies with one Ultra power.
Side-tracked (10 points): Complete your first challenge mission.
Taking Every Opportunity (25 points): Complete 10 unique challenge missions.
Completionist (50 points): Complete all 15 unique challenge missions.
Garbage Collection (10 points): Destroy your first piece of propaganda.
Cleaned up the City (30 points): Destroy 25 pieces of propaganda.
Profiler (10 points): Collect your first dossier.
Archivist (30 points): Collect 15 dossiers.
Purify the Purifiers (20 points): Defeat 2,000 Purifiers.
Mechageddon (20 points): Defeat 20 Purifier Stalker Mechs.
Reinforced (20 points): Defeat 20 Prime Enforcers.
U Mad, Bro? (20 points): Defeat 30 U-Men in the Secret Lab.
The Goon Squad (20 points): Defeat 500 MRD Troops.
Alpha Level Mutant (50 points): Finish the game on X-Man difficulty.
Omega Level Mutant (50 points): Finish the game on X-treme difficulty.
Can I Get A Valkyrie? (0 points): You were defeated 100 times.
Beta Level Mutant (50 points): Finish the game on New Mutant difficulty.

Additionally, there are 18 secret achievements:

Teleport This! (15 points): Help Nightcrawler rescue mutants.
Stay Frosty (15 points): Help Iceman, Cyclops and Emma Frost defeat the Purifiers.
Flash Fire (15 points): Help Pyro, Juggernaut and Quicksilver defeat the Purifiers.
Satellite Interference (15 points): Interrupt the transmission.
Better than the Best (15 points): Defeat more enemies than Wolverine in the Prime Enforcer factory.
The Roof, the Roof... (15 points): Help Pyro set the roof on fire.
Why Do They Keep Coming? (15 points): Help Forge's hacked Purifier Stalker Mech defeat the MRD Troops.
Mutant Tracker (15 points): Rescue Caliban.
Another Shrimp on the Barbie (15 points): Stop Pyro from being mind-controlled.
Fist of the... (15 points): Stop Northstar from being mind-controlled.
Logan's Run (15 points): Help Emma Frost turn Wolverine back to normal.
It's a Secret to Everybody (15 points): Find the U-Men secret lab.
Shock and Awe (15 points): Defeat Cameron Hodge.
An Unstoppable Force (15 points): Defeat Magneto and Juggernaut.
Diamond in the Rough (15 points): Help Emma Frost defend mutant civilians from Purifier attacks.
Things Look So Bad Everywhere... (15 points): Survive Magneto's attack on the Purifiers.
Ace In The Hole (15 points): Defeat Gambit.
David Beats Goliath (15 points): Defeat Sublime.