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