Here is the full list of all 38 House of the Dying Sun achievements.
Earn a second wingman.
Earn a frigate for your fleet.
Assassinate Commodore Freyv.
Assassinate Lord Commodore Ganuun.
Destroy a vessel while inside its shields.
Add a destroyer to your fleet.
Earn a wingman.
Earn a new weapon.
Purchase an upgrade.
Complete a bonus objective.
Complete a mission on Invader or higher.
Complete a mission on Warlord or higher.
Complete a mission on Harbinger.
Destroy an escape pod.
Complete host uplink training.
Purchase all upgrades.
Run over a traitor lord's escape pod.
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 Ausgauth.
Assassinate Commodore Elduette.
Assassinate Lord Commodore Seigfast.
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.