Here is the full list of all 20 The Solitaire Conspiracy achievements. It takes around 4-5 hours to unlock all of the achievements on Windows.
Complete Level 4
Complete Level 3
Complete Level 2
Change the game's color theme for the first time
Complete Level 1
Complete Level 5
Complete Level 6
Complete Level 7
Complete Level 8
Complete Level 9
Complete Level 10
Complete Level 11
Complete Level 12
Complete Level 13
Complete the Campaign
Get to Wave 5 in Countdown
Get to Wave 10 in Countdown
Get to Wave 15 in Countdown
Complete a Skirmish with only one crew
Complete a Skirmish with four crews