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