To some extent, the Klingon monoculture must be an effect simply of the types of Klingons shown -to tell a particular story at a particular time. Whenever a warship is shown, there will of course be warriors aboard. When we are shown Klingon leaders, they are all the heads of the ruling families (houses), who seem to operate to some extent like feudal barons of the early middle-ages. The fact that they have a functioning star navy indicates that there must be engineers and builders. A top Klingon chief engineer will surely be just as good an engineer as LaForge or O'Brien, but probably a bit more handy in combat. When a Klingon house is dishonoured, they may lose their lands and wealth, which surely implies that beneathe the ruling class they have a serf population who work those lands and create the produce and wealth, that the nobles just cream off (much like middle age lords).
I would also assume that the Klingons would employ mandatory 'national service' of those who have come of age too (as ranks), and that traditionally, offspring of the ruling houses would become warriors (as officers)- no different from military tradition in some upper class families today. In short, we hardly ever see a non-military Klingon, because Trek writers didn't see an interesting way to use non-military Klingons much.
Which non military Klingons do we see? there's the tribunals that send Kirk and Archer to Rura penthe, Archer's lawyer, Lady Grilka (OK she's the widow of a warrior), the DS9 chef, the inhabitants of Carreya IV (but they are effectively independant) the Duras sisters seem to be renegade ex-warriors but that's about it. all other Klingons we see are warriors, or associated with the ruling houses, which would make them ex-warriors - because these are the Klingons who have contact with Starfleet. Given his status as a minor noble, Worf would have been yet another 'nameless' warrior had he not been orphaned and taken in by the Rozhenko's.
It would have been interesting to meet a Klingon freighter captain, or perhaps merchants passing through DS9 on their way to the Gamma Quadrant to negotiate trade deals before the Dominion War, or even a Klingon engineering team getting warp drive going again after battle damage.
Certainly the Klingon economy did work around money, unlike the Federation, so there would probably be all the 'white-collar' city workers too - but that does not make for an interesting trek story. Who wants to see a Klingon farmer ploughing a field, or a Klingon plumber fixing a bust pipe?