I have a question: We know of Klingon - Kinshaya relations. What do we know about relations between the Klingons and the other members of the Typhon Pact?
They have some sort of dispute with the Gorn over territory ("Mere Mortals" established, if I remember correctly, eight star systems the Gorn thought were by rights theirs, but the Klingons had control over). There is no indication it is a violent dispute, though. In "Serpents Among the Ruins", the Klingons are sending fleets to patrol the Gorn border, but don't seem to think there'll be trouble. I don't think the Gorn and Klingons have truly fought; indeed, in "The Gorn Crisis" they seemed to get along quite well following the Black Crest defeat. Ambassador Zogazin's comments in "Mere Mortals" suggest the Gorn see the Klingons as a Federation puppet state, though.
As we all know, they hate the Romulan Star Empire due to the endless stream of honour disputes, blood debts and what it insists is the Romulan's cowardly betrayal. They've never actually been at war, as far as I can tell, at least not since the 22nd century, but each would love to see the other fall. Amusingly, the Klingons seem to have no problem with the Imperial Romulan State, so perhaps hatred for Romulans-the-species is being transformed into more direct hatred for the RSE.
They have diplomatic relations with the Tzenkethi, but I don't think there has been anything of note between them.
They're hostile to the Tholians, as "Vanguard" and several "SCE" stories have shown, but in the latter Worf managed to prevent a conflict starting, so there appear to be plenty of heads cool enough to put aside lingering grudges over the Taurus Reach stuff. Note: did anyone else catch the link between "Open Secrets" and those SCE stories?
As for the Breen, there was Emperor Mow'ga (is that right?) and his long-ago attempt to conquer them, which failed spectacularly, and of course the Dominion War, but other than that, I think they're too far apart to have had any real contact.