Here is the full list of all 38 House of the Dying Sun achievements.
Earn a frigate for your fleet.
Run over a traitor lord's escape pod.
Add a destroyer to your fleet.
Destroy a vessel while inside its shields.
Earn a new weapon.
Earn a wingman.
Complete a mission on Harbinger.
Destroy an escape pod.
Complete a mission on Warlord or higher.
Purchase an upgrade.
Complete a bonus objective.
Complete a mission on Invader or higher.
Complete host uplink training.
Earn a second wingman.
Purchase all upgrades.
So I could rest.
To avenge the Emperor.
To free the galaxy.
Because I love to kill.
Because this isn't real.
Because I am not human.
Complete at least 1 battle in Challenge Mode
Survive for at least 5 battles in Challenge Mode.
Survive for at least 10 battles in Challenge Mode.
Assassinate Commodore Freyv.
Assassinate Commodore Ausgauth.
Assassinate Commodore Elduette.
Assassinate Lord Commodore Seigfast.
Assassinate Lord Commodore Ganuun.
Assassinate Lord Commodore Naybard.
Assassinate Traitor Admiral Anrune.
Assassinate Traitor Admiral Bergautur.
Assassinate Traitor Admiral Alfmundur.
Assassinate Prince Aum.
Assassinate Princess Admiral Du Moch.
Assassinate Kingsclone A1.
Assassinate Kingsclone B1.
Assassinate The Traitor King.