「自分{じぶん}がどんな人{ひと}であるかを知{し}ってもらいたい。」 ("I want you/them/people, etc. to know what kind of person I am.")
This is a completely grammatical and natural-sounding sentence, including the use of 「を」.
Why can を act on a particle? I understand のか, なんか as they are compound words which can be treated as nouns, i.e direct objects by を (or so I believe), but why here?
The 「を」 does not act only on the particle 「か」 here. It is acting on the whole preceding phrase 「自分がどんな人であるか」, which is a noun phrase.
「noun phrase + を + 知る」 is perfectly correct. It does not have to be a simple single-word noun that immediately precededs the 「を」.
「どんな人であるか」 means virtually the same thing as 「どんな人であるのか」 and both are noun phrases.
Is it simply omitting the の because it's obvious (のかを)→(かを), and if not what's the difference between のか and か in that sentence?
I would not say it is omitting the 「の」 because it is already "complete" without it. This 「の」, if used, helps emphasize the statement. It would be, in a way, analogous to the difference between "what kind of person I am" and "what kind of person I really am".