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