Here is the full list of all 27 Star Dynasties achievements.
Repair your first installation
Win a tournament
Trigger your first expedition event
Assign a house member to a task
Have all places in your council filled
Customize your character
Complete the tutorial
Win a battle
Complete 10 turns in one game
Complete 20 turns in one game
Conquer your first system
Have 10 house members assigned to a task
Press a claim
Have a league of 15 systems
Have a league of 30 systems
Host a feast in someone's honor
Have an affair
Win the game
Have 250 money
Have all house scores above 200
Have a baron swear allegiance to you
Have a duke swear allegiance to you
Fully acquire a bloodline upgrades branch
Have 3 of your children be wards of foreign rulers simultaneously
Raise a child from birth and have them come of age with a skill greater than 50
Design a family of 16 characters
Have a foreign ruler inherit your house