Here is the full list of all 38 House of the Dying Sun achievements.
Complete at least 1 battle in Challenge Mode
Complete a bonus objective.
Complete a mission on Invader or higher.
Complete host uplink training.
Earn a wingman.
Add a destroyer to your fleet.
Earn a frigate for your fleet.
Complete a mission on Warlord or higher.
Complete a mission on Harbinger.
Destroy a vessel while inside its shields.
Purchase an upgrade.
Earn a new weapon.
Earn a second wingman.
Purchase all upgrades.
Destroy an escape pod.
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 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.