We all know that the Klingon faction, in its current form, is not the equal of the Federation faction in terms of content. Most STO forum denizens seem to agree that Klingons need more content, but a part of the issue is that the content cannot simply be re-skinned Federation content.
So, how would Klingon diplomacy work? Klingons rule an empire and are regarded as one of the preeminent military powers in the Alpha Quadrant. Klingons are guided by a sense of honor and in general seem to have a more warlike worldview than the Federation.
We've never heard too many great details about the demographic structure of the Klingon Empire; we do know, in STO, that the Empire consists of Klingon-populated worlds, Gorn worlds, and likely Orion and Naussican worlds. There are also the Klingon Great Houses and many other internal political forces to deal with.
I envision Klingon diplomacy as having a far more substantial internal component than Federation diplomacy.
We already have the Great Houses. We could choose a handful of canonical Great Houses and bring them to life in the game. Klingon players could choose to work for a particular great house and deal with that houses's diplomatic concerns on the Homeworld and on vassal worlds. A Klingon diplomat would need to balance both his House's own desire for honor and prestige on the Homeworld with the external goals of that house.
Internal conflict appears to be the key to authentic Klingon diplomacy, and I think there is enough background material here to make Klingon diplomacy have a very uniquely Klingon feel to it.