ooooo I am Soooooo tempted to say something funny, but alas I shall endeavor to answer correctly LOL
The best easiest straight answer is to make level 6 as a Fed. That will auto unlock the ability to make a Klingon character (provided you have the character slot to do so)
It used to be 6, then it was 5, now it's 6 again.
There is a quest or 2 you can complete that will also auto unlock the option.
One that I am aware of off the top of my bumpy head is Silence the Echo or something like that.
If I have misnamed it I am sure someone will correct me but if you have the quest in your log you will know it from my referrence.