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
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.
Because I am not human.
Because this isn't real.
Because I love to kill.
To free the galaxy.
Purchase all upgrades.
So I could rest.
Assassinate The Traitor King.
Destroy a vessel while inside its shields.
Complete a mission on Harbinger.
To avenge the Emperor.
Earn a second wingman.
Earn a frigate for your fleet.
Add a destroyer to your fleet.
Run over a traitor lord's escape pod.
Purchase an upgrade.
Complete a mission on Warlord or higher.
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.