As everyone knows the exchange is not put in on the Klingon side.
For a Klingon to use the exchange you will need help of 1 friend.
Have your friend go to his bridge and invite you to his bridge. Once you are there you can trade items or credits to your friend.
Next log off and log on to your Fed toon (lvl 4-6 that was used to unlock Klingons). Have your friend invite this toon to his bridge. YOU MUST STAY IN THE BACK OF THE BRIDGE as his bridge officers will attack if you get close. He then can trade with you as a Fed toon.
So the idea is if your a Klingon you pass some credits to your Fed toon and buy stuff on the Exchange and then pass it back to your Klingon.
Hope this made sense.
EDIT - Forgot to mention that it is much easier if you have your friend "friend" you.