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There are two options for unlocking the ability to create a KDF aligned Klingon Character within Star Trek Online:
(1) Advance your Federation character to Liuetenant Grade 6 (Level 6), at which point the option for Klingon play will unlock.
(2) Run missions with your Federation character until you are able to do the mission "Stop the Signal", which also unlocks Klingon play when completed. This method is available as early as Liuetenant Grade 4 (Level 4) if you follow this guide, and is the fastest known method for unlocking the option to create a Klingon character.
Option 2 sounds intriguing doesn't it? What is the quickest way to achieve this, you ask?
Here is a run-down of the missions that lead to "Stop the Signal", including an estimate of the time required to complete each mission (solo play):
Tutorial - 15 minutes
SS Azura - 15 minutes
Diplomatic Mission - 20 minutes
Hide and Seek - 30 minutes
SB 24 Fleet Action - 10 minutes
SB 24 Ground Mission - 25 minutes
Stop the Signal - 20 minutes (This mission unlocks Klingons when you finish it, sometimes requires a relog to take effect, or if you have unspent skill points you need to spend them and then the faction will unlock for play).
It requires approximately 2 hours and 20 minutes of time to achieve this goal. I have noticed alot of posters seem unaware that you are able to unlock using the method described above, and seem to think you are forced to wait until reaching L6, which is not the case. If you want to unlock the Klingons as fast as possible, this is the easiest and quickest method available.
Hopefully this helps some of the newer players and anyone in general who was unaware of this method for unlocking