there is less klingon content than fed (but i guess you know that).
good news is you start at level 6, and the first few levels up to lieutenant are pretty easy. you have the devidian series starts at level 6, the breen at level 10, and a new series starting this saturday.
klingon sorties are located in pi canis sector. these can be repeated every half an hour. pretty simple missions but quite fun. there is an optional side objective mission to each one that gives further rewards if you take and complete the path of the warrior mission. there are more sorties due.
you can now do deep space encounters in psi vallorum.
there are 8 unique klingon episode missions (2 at level 10. 2 at level 20, 2 at level 30 and 2 at level 31) these are great eps so make sure you play them.
there are lots of little missions to do that you should always try and take when available. like destroy 10 turrets, or kill 50 enemies, or win 3 pvp matches.
try to pvp as much as you can. arena matches in space are usually the most fun.
defend the empire missions are the equivalent to deep space encounters but you have to go to them as they are fixed locations on the map.
you may have to grind a lot of starclusters or do the repeatable patrols in eta and orellius sector to level to compensate for the lack of pvp.
get the 8 hours 20% bonus from the q mission at qo'nos now and use it when you feel its the right time.
try to stick to bop, they will offer unique gameplay options like universal bridge slots and battle cloaks.
take it slowly to let the devs catch up on content and try to use the foundry when it comes out. because of the lack of content dont burn yourself out to much as it can be a grind.
not sure if that helps at all. its more of less the same as feds once you get used to it.
I'll also add that from Commander to Captain, I did the Deferi dailies as often as possible. I would do these, combined with the Satellite daily and the Fuel Depot daily in Eta Eridani. One set of these would normally clear me through one third to one half of a level.
Also get yourself into PvP arena queues and be prepared to drop whatever you're doing to join in. Space are the most fun, but some of the ground ones can be interesting.
As the Cap. Revo mentioned there are a number of 'side missions' which are sort of like 'achievement missions.' Kill a total of N turrets (space and ground both count towards this!), kill X NPC ships, kill N enemy captains, participate in N arena battles.
When you get to Gamma Orionis, the best place to grind levels is Borg DSEs. Not only does the XP come in fairly steadily from the 'Defend Gamma O' repeatable, but they're a lot of fun with upwards of 20 other players blowing Borg up. Gamma O also gives access to the Undine Front missions which are available to both sides.
You'll also get to do the Borg STFs which... are kind of the last great outpost of misery for Klingons. There are very few Klingons doing STFs at a given time, making getting a Klink team together kind of rough. Just so you know, it's a little rough if you're looking forward to STFs.