Here is the full list of all 27 Star Dynasties achievements.
Complete 20 turns in one game
Repair your first installation
Have all places in your council filled
Complete 10 turns in one game
Trigger your first expedition event
Conquer your first system
Win a battle
Have a baron swear allegiance to you
Assign a house member to a task
Customize your character
Complete the tutorial
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 a tournament
Win the game
Have 250 money
Have all house scores above 200
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