Here is the full list of all 38 House of the Dying Sun achievements.
Survive for at least 5 battles in Challenge Mode.
Complete at least 1 battle in Challenge Mode
Add a destroyer to your fleet.
Purchase an upgrade.
Destroy a vessel while inside its shields.
Complete a mission on Warlord or higher.
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.
Earn a frigate for your fleet.
Complete a mission on Harbinger.
Earn a second wingman.
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.
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.