Here is the full list of all 38 House of the Dying Sun achievements.
Assassinate Commodore Elduette.
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
To avenge the Emperor.
Assassinate The Traitor King.
Assassinate Kingsclone B1.
Complete a mission on Warlord or higher.
Complete a mission on Harbinger.
Earn a second wingman.
Earn a frigate for your fleet.
Purchase an upgrade.
Add a destroyer to your fleet.
Earn a wingman.
Earn a new weapon.
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.
Run over a traitor lord's escape pod.
So I could rest.
To free the galaxy.
Because I love to kill.
Because this isn't real.
Because I am not human.
Assassinate Commodore Freyv.
Assassinate Commodore Ausgauth.
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.