I think a hypothetical Klingon show can't be just
about the Klingons. If anything, it should probably show that the Klingon Empire actually consists of many
races like the Federation, but that Klingons naturally dominate. Some of those races may have been conquered, while others may have willingly joined the Klingons for various reasons (including a dislike of the Federation).
It should paint the Klingons as being more than one-dimensional, perhaps even revealing that most of their warrior bluster is more of a long-standing policy with other nations and that some hidden non-warrior aspects of the Empire would surprise us, including the roles of non-Klingons within it.