You'll unlock this trophy by completing the Klingon Empire's second story arc, "Empire." You can check your progress in-game for each story arc by pressing Options > Missions > Missions Journal > Episodes. This includes the following episodes/missions:
- Duties of Command
- The Hunt is On
- Space Chase
- Manhunt
- Friend or Foe
In order to access these episodes/mission you have to make a character from the Klingon faction, race isn't important.