Here is the full list of all 74 Crash Bandicoot N. Sane Trilogy achievements. It takes around 40-50 hours to unlock all of the achievements on Windows.
Do hard time
Just "forget" it. Repeatedly.
Don't feel guilty about jumping on him.
Earn 27 Relics (gold or better)
Earn 42 Gems
Unlocked the Secret Exit in Unbearable
Earn 99 Lives
Defeat 5 enemies with Coco's Double Leg Drop
Earn 10 Relics (gold or better)
Earn 26 Relics (gold or better)
Discover the Secret Exit in Diggin' It
Discover the Secret Exit in Bear Down
Discover the Secret Exit in Air Crash
Earn 26 Gems
Seek and destroy
Earn the Key in Jaws of Darkness
Earn the Key in Sunset Vista
Seek and destroy. Again.
Shoot the Imposter
Earn 30 Relics (gold or better)
Have your bazooka ready.
Shoot 5 of them in a single level.
Avoid whatever he's flinging at you.
Die.
Earn 13 Gems
Defeat 5 enemies with Coco's Spin
Defeat N. Cortex
Defeat N. Gin
Earn 21 Gems
Unlocked the Secret Exit in Hangin' Out
Defeat N. Brio
Discover a Gem Path after earning a Color Gem
Die as Crash.
Earn 5 Relics
Spin away an Extra Life
Defeat Pinstripe Potoroo
Earn the Gems in the bonus levels
Discovered the Secret Exit in Dino Might!
Discovered the Secret Exit in Road Crash
Defeat Tiny Tiger
Defeat the Komodo Brothers
Fly, then die.
Discover a Death Route
Defeat Koala Kong
Defeat Ripper Roo
Discover a "?" Bonus Path
Miss a lot of boxes.
Shoot a background UFO in Future Tense.
Defeat 5 enemies with Coco's Savage Hurricane Spin
Defeat N. Tropy
Defeat Dingodile
Defeat Papu-Papu
Enter a Bonus Round after collecting 3 Character Tokens
In a single level, spin an enemy into another enemy twice