Here is the full list of all 38 House of the Dying Sun achievements.
Assassinate Traitor Admiral Alfmundur.
Assassinate Lord Commodore Seigfast.
Assassinate Traitor Admiral Anrune.
Assassinate Commodore Ausgauth.
Assassinate Commodore Freyv.
Run over a traitor lord's escape pod.
Assassinate Commodore Elduette.
To avenge the Emperor.
Earn a second wingman.
Complete a mission on Harbinger.
Complete a mission on Warlord or higher.
Earn a frigate for your fleet.
Add a destroyer to your fleet.
Purchase an upgrade.
Earn a new weapon.
Earn a wingman.
Destroy a vessel while inside its shields.
Complete a bonus objective.
Complete a mission on Invader or higher.
Destroy an escape pod.
Complete host uplink training.
Purchase all upgrades.
So I could rest.
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 Lord Commodore Ganuun.
Assassinate Lord Commodore Naybard.
Assassinate Traitor Admiral Bergautur.
Assassinate Prince Aum.
Assassinate Princess Admiral Du Moch.
Assassinate Kingsclone A1.
Assassinate Kingsclone B1.
Assassinate The Traitor King.