Here is the full list of all 25 Halo: Spartan Assault achievements. It takes around 15-20 hours to unlock all of the achievements on Windows.
Complete all campaign mission specific Assault Ops in campaign A to E
Finish the objective without using any vehicles/turrets in Operation E, Mission 4
Complete all Operation F mission Assault Ops
Complete Operation F, Mission 4 with the Blind and Thunderstorm Skulls active
Complete Operation E, Mission 1 with the Hollow and Black Eye Skulls active
Earn all Gold stars in Operation F
Earn all Gold stars in all operation missions
Finish the objectives using only the Brute Shot, melee and grenades in Operation D, Mission 1
Kill a convoy target with every stationary gun in Operation C, Mission 3
Score a kill with every handheld weapon in the campaign
Finish the mission without any Jackal regenerating its shield in Operation B, Mission 3
Earn 25,000 XP throughout your career
Earn all medals in the campaign at least once
Use every ability in the campaign at least once
Complete Operation E
Complete Operation F
Destroy all Mega Turrets with the Grizzly tank in Operation F, Mission 2
Complete Operation D
Complete any campaign mission with the maximum number of Skulls active
Complete Operation C
Complete Operation B
Complete Operation A
Keep all Wolverines intact in Operation A, Mission 3
Earn your first Gold star in an operation mission
Earn a star of any type in an operation mission