Level a Fed character to Lt/6 to unlock Klingons and a 3rd slot (for either faction) opens.
During the pre-order period, a 12-month or lifetime subscription included 2 additional slots over the 3 that normal accounts get. Those offers ended on the 1st, however.
It is expected that extra slots will become available from the Cryptic store at some later date. I would also expect that expansions with additional factions (Romulans, Cardassians, Dominion) will come with more slots, but I could be wrong on that.