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
Complete a mission on Harbinger.
Complete a mission on Warlord or higher.
To free the galaxy.
Earn a second wingman.
Earn a frigate for your fleet.
Run over a traitor lord's escape pod.
Purchase an upgrade.
Add a destroyer to your fleet.
Destroy an escape pod.
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.
Complete host uplink training.
Purchase all upgrades.
So I could rest.
To avenge the Emperor.
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.