Here is the full list of all 38 House of the Dying Sun achievements.
To avenge the Emperor.
Earn a second wingman.
Earn a frigate for your fleet.
Purchase an upgrade.
Complete a mission on Harbinger.
Complete a mission on Warlord or higher.
Add a destroyer to your fleet.
Destroy a vessel while inside its shields.
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.
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.
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 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.