There's a few bugs, but they should be fixed soon.
You access the Klingon Academy Zone by, reaching Level 5 (I think?) which is Lt. 3 (I think?) and then you create a Klingon Character play through the tutorial; then head to the 'transporter room', stand on a transporter pad, and tell the transporter chief where you want to go.
N.B. The klingon academy zone is ONLY available on the Tribble Test-Shard at time of posting.