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