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.
Score a kill with every handheld weapon in the campaign
Earn all medals in the campaign at least once
Finish the objectives using only the Brute Shot, melee and grenades in Operation D, Mission 1
Complete Operation F
Use every ability in the campaign at least once
Destroy all Mega Turrets with the Grizzly tank in Operation F, Mission 2
Complete Operation E
Complete Operation D
Complete Operation C
Kill a convoy target with every stationary gun in Operation C, Mission 3
Complete Operation B
Finish the mission without any Jackal regenerating its shield in Operation B, Mission 3
Complete Operation A
Complete any campaign mission with the maximum number of Skulls active
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
Finish the objective without using any vehicles/turrets in Operation E, Mission 4
Earn all Gold stars in all operation missions
Earn 25,000 XP throughout your career
Complete Operation E, Mission 1 with the Hollow and Black Eye Skulls active
Complete all campaign mission specific Assault Ops in campaign A to E
Earn all Gold stars in Operation F
Complete Operation F, Mission 4 with the Blind and Thunderstorm Skulls active
Complete all Operation F mission Assault Ops