And when they say Klingon they don't necessarily mean Klingon like Worf and Martok -- there are several species (and create your own species) that are including in the Klingon faction. Gorn and Orion come immediately to mind but their are others.
As you'll soon discover STO is a very incomplete game. Once you hit level 6 you'll find there's a lot more to do pvewise with the federation then the klingons. I suspect they did the level 6 thing so that noobs have an easier time leveling and learnig initially.
Bottom line, near as I can tell, Cryptic was contractually obligated to have this game out by a certain date; the previous 2 companies that were working on this game poofed into vaporware and if Cryptic didn't deliver then it would have been the final nail in STO's coffin. So, they released on time, but with a severely <ahem> shortened game. They're adding more content, and their plans is to make the Klingons a full faction like the Feds are, as well as (eventually) the Romulans and Cardassians I've heard.
Yeah. Currently the Klingons are pretty much PvP only. There's some token repeatable PvE combat encounters for the Klingons, but no true story content.
That said, we will be getting Exploration Missions (Genesis stuff) on the Klingon side soon - that's confirmed on the main webpage and in dev posts here. They're aiming to release this first batch of content updates every couple weeks or so - I doubt they'll be able to sustain that pace, but we should get our Exploration content soon.
The devs have SAID they WANT to add full episodic PvE story content for the Klingons, but we have no official ETA on this.
Future factions like Romulans and Cardassians are simply a possibility, nothing more - they're a strong possibility, but that's all the devs have acknowledged. It's not been promised as a certainty.