I am confused translating "klingonness" (Quality of being Klingon, akin to Humanity). I understand that there is no such thing as an abstract noun affix from a noun in Klingon and that tlhIngan is a noun and not a adjectival verb (i.e. to be klingon –I find it odd there isn't), so *tlhInganghach is wrong.
tlhIngan Dotlh (Klingon status) seems lacklustre and tlhingan chaHghach (klingon being) is most likely wrong, but I am not familiar enough with Klingon to think of a noun that fits the bill.
It's very difficult to translate single words between languages, because different languages express the same concepts in different ways. It would be much better if you can provide some context for why you want to translate this concept. For example, are you translating a sentence in which someone speaks about someone's "Klingonness"?