As per this (cited by "An Analysis of Dwarvish" in Arda Philology 1: Proceedings of the First International Conference, with a very nice glossary of the language, which can be found here):
Regarding Khuzdul, Tolkien stated that "this tongue has been sketched in some detail of structure, if with a very small vocabulary" (PM:300). It evidently came into being in the thirties.
Here PM refers to the Peoples of Middle Earth. The complete corpus of the language is described on the same site as being
a few names, like Khazad-dûm and Zirak-zigil, the inscription on Balin's tomb, and a battle cry: Baruk Khazâd! Khazâd ai-mênu! "Axes of the Dwarves! The Dwarves are upon you!"
A petty-dwarf, Mîm, is also named, in The Children of Hurin, though that is the only reference there. The names also include Chamber of Mazarbul [records].
(A complete glossary is given on the website; it is too long to copy here.) It also mentions Appendix D, E, and F in the Lord of the Rings (F primarily containing history, of course) and the Appendix in the Silmarillion also apparently contains a reference to Khuzdul:
At least one Khuzdul word made its way into Sindarin: kheled "glass", that appears in Grey-elven as heledh (see the Silmarillion Appendix, entry khelek-)
This appears to be all we know of Khuzdul. In spite of this, extensive analysis has been made, with a not insignificant amount of it guesswork. See "An Analysis of Dwarvish" for one example (provides a theory of language structure).
Of course, Tolkien did state that it was "sketched", but the question is where it was - there's a chance it may be unpublished:
Yes, there is some unpublished material concerning Khuzdul. However, it’s only a small amount, and as I recall, it consists mostly of phonological discussion, though inevitably there is in conjunction with this some discussion of the grammatical significance of various root modifications. But don’t expect anything like Etymologies or the "Early Qenya Grammar" for Khuzdul.