Here is the full list of all 20 Halo: Spartan Strike achievements. It takes around 12-15 hours to unlock all of the achievements on Windows.
Complete Operation A
Complete Operation A, Mission 6 with the Blind and Thunderstorm Skulls active
Keep the Elephant 100% intact in Operation A, Mission 3
Earn your first Gold star
Complete Operation B
Complete Operation C
Complete Operation D
Complete Operation E
Keep the Wraith 100% intact in Operation B, Mission 4
Do not allow the Watchers to resurrect any Knights in Operation C, Mission 4
Complete Operation D, Mission 5 without getting hit by a Focus Turret
Do not get hit by any Proximity Mines in Operation E, Mission 2
Earn all Gold stars in Operations A through E
Spend 10,000 credits on loadout upgrades
Use every Armor Ability at least once
Complete Operation B, Mission 6 with the Hollow and Pacifist Skulls active
Complete Operation C, Mission 6 with the Black Eye and Pacifist Skulls active
Complete Operation D, Mission 6 with the Famine and Thunderstorm Skulls active
Complete Operation E, Mission 6 with the Hollow and Black Eye Skulls active
Complete all Assault Ops for Operations A through E