Here is the full list of all 38 House of the Dying Sun achievements.
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
Because I love to kill.
To free the galaxy.
To avenge the Emperor.
Because I am not human.
Purchase all upgrades.
Because this isn't real.
Assassinate The Traitor King.
Assassinate Kingsclone B1.
Assassinate Kingsclone A1.
Assassinate Princess Admiral Du Moch.
Assassinate Prince Aum.
Assassinate Traitor Admiral Alfmundur.
Assassinate Traitor Admiral Bergautur.
Assassinate Lord Commodore Naybard.
Assassinate Lord Commodore Ganuun.
Assassinate Lord Commodore Seigfast.
Assassinate Traitor Admiral Anrune.
Assassinate Commodore Elduette.
Assassinate Commodore Ausgauth.
Assassinate Commodore Freyv.
Complete a mission on Harbinger.
Complete a mission on Warlord or higher.
So I could rest.
Earn a second wingman.
Earn a frigate for your fleet.
Purchase an upgrade.
Add a destroyer to your fleet.
Destroy a vessel while inside its shields.
Run over a traitor lord's escape pod.
Earn a new weapon.
Earn a wingman.
Complete a bonus objective.
Complete a mission on Invader or higher.
Destroy an escape pod.
Complete host uplink training.