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