Here is the full list of all 128 Duke of Alpha Centauri achievements. It takes around 2-3 hours to unlock all of the achievements on Windows.
Destroy your ship 10 times.
Spend 6 ship upgrade points.
Destroy your ship 7 times.
Use fast repair 34 times.
Spend 5 ship upgrade points.
Spend 50 system upgrade points.
Reach player level 50.
Reach player level 45.
Survive one minute in endless mode.
Survive 3 minutes in endless mode.
Survive 5 minutes in endless mode.
Survive 2 minutes in endless mode.
Survive 4 minutes in endless mode.
Destroy your ship 5 times.
Play endless level.
Spend 40 system upgrade points.
Finish level 20.
Finish all levels on any difficulty.
Kill Proxima.
Reach player level 40.
Use fast repair 21 times.
Destroy your ship 3 times.
Spend 4 ship upgrade points.
Finish level 19.
Destroy ship type 2675 Tolkien.
Reach player level 35.
Finish level 18.
Use holodrones 29 times.
Spend 3 ship upgrade points.
Spend 30 system upgrade points.
Reach player level 30.
Finish level 17.
Finish level 16.
Reach player level 25.
Finish level 15.
Use holodrones 23 times.
Use fast repair 13 times.
Destroy ship type 99942 Apophis.
Finish level 14.
Destroy ship type 1235 Schorria.
Use holodrones 17 times.
Spend 2 ship upgrade points.
Spend 20 system upgrade points.
Reach player level 20.
Finish level 13.
Destroy ship type 4524 Barklajdetolli.
Destroy ship type 1036 Ganymed.
Destroy 100 enemy waves.
Finish level 12.
Use fast repair 8 times.
Destroy ship type 912 Maritima.
Destroy missile type 18 Melpomene.
Reach player level 15.
Finish level 11.
Destroy ship type 9165 Raup.
Use holodrones 11 times.
Use fast repair 5 times.
Destroy ship type 192 Nausikaa.
Finish level 10.
Kill Rigil.
Use fast repair 3 times.
Use holodrones 7 times.
Use fast repair 2 times.
Destroy 250 asteroids.
Spend ship upgrade point.
Spend 10 system upgrade points.
Reach player level 10.
Finish level 9.
Shoot 250 rockets.
Finish level 8.
Destroy 500 enemy ships.
Destroy 50 enemy waves.
Destroy missile type 15 Eunomia.
Finish level 7.
Destroy ship type 433 Eros.
Destroy 100 asteroids.
Destroy 250 enemy ships.
Use holodrones 3 times.
Reach player level 5.
Finish level 6.
Destroy ship type 846 Lipperta.
Use fast repair.
Shoot 100 rockets.
Shoot 1000 plasma bullets.
Finish level 5.
Destroy 25 enemy waves.
Destroy 50 asteroids.
Use holodrones.
Destroy 100 enemy ships.
Shoot 500 plasma bullets.
Shoot 50 rockets.
Finish level 4.
Shoot 250 plasma bullets.
Finish level 3.
Destroy 25 asteroids.
Destroy 50 enemy ships.
Shoot 25 rockets.
Destroy ship type 2000 YY.
Shoot 100 plasma bullets.
Spend system upgrade point.
Level up!
Finish level 2.
Shoot 1000 bullets.
Shoot 10 rockets.
Destroy 10 asteroids.
Destroy 10 enemy waves.
Destroy ship type 1999 XC.
Shoot 50 plasma bullets.
Finish level 1.
Shoot one rocket.
Shoot 500 bullets.
Destroy 10 enemy ships.
Shoot 250 bullets.
Shoot 25 plasma bullets.
Shoot 10 plasma bullets.
Destroy asteroid.
Shoot 100 bullets.
Shoot plasma bullet.
Shoot 50 bullets.
Destroy ship type 7 Iris.
Shoot 25 bullets.
Shoot 10 bullets.
Shoot bullet.
Finish tutorial level.
Destroy enemy wave.
Destroy enemy ship.
Destroy ship type 288 Glauke.
Destroy your ship.