Starfleet is to the Federation as what is to the Klingon Empire? I'm looking for the organisation that the Klingon Empire has which serves a similar function to Starfleet. I thought perhaps the Klingon Defence Force, but they seem to be a very militaristic organisation (that being said, what isn't militaristic about the Klingons...) whereas Starfleet, in addition to their militaristic role, also has roles in exploring and contacting new life. So, what is the Klingon Empires' equivalent of Starfleet?
In short, no, there's no direct Klingon equivalent of Starfleet
...that is, if you believe that Starfleet is an exploration and peacekeeping force exclusively. This is what they claim.
But they are significant arguments that Starfleet is also a military organization:
So it depends if you go by what Starfleet says it is, or what role(s) it actually performs.
If you consider Starfleet's military role to be signficant, then the Klingon equivalent is The Klingon Defense Force.
With regard to the peacekeeping: To the extent this is possibly a euphemism for military, then the Klingons would say the KDF performs that function. (You can keep the peace by brute force, occupation and annexation.)
With regard to exploration: There appears to be no body performing strict exploration, although, for example, KDF ships do go through the wormhole in DS9. One may be able to argue about whether the exploration is for the purpose of military advancement or not, and whether it counts as exploration for the purpose of exploration or not if the motivation is military.