How did those unworthy Terran petaQ manage to plagiarize Shakespeare so many years before first contact?

Lucky you, OP, because I wrote the introduction to The Klingon Hamlet, and translated the verse of the play (or rather, in-universe, I was editorially involved in the Terran edition of the play Tragedy of Khamlet, Son of the Emperor of Kronos, by Wil’yam Shex’pir, and translated the introduction).

And the introduction pays glancing mention to this issue. If I may quote myself (or rather, in-universe, what I translated):

It is regrettable that, during the years when the Empire and Federation were at war—a war the Federation fought on the propaganda front even more keenly than on the battlefield—certain indivduals resorted to crude forgeries of Shex’pir, claiming him as a conveniently remote mediaeval Terran, a certain Willem Shekispeore, and hoping by this falsification of history to discredit the achievements of Klingon culture.

We will not dwell on this unfortunate episode, although we are dismayed by the fact that this belief persists amongst many in the Federation to this day. In this edition, we juxtapose the Klingon original with the most prevalent of the versions of “Amlet” purported to have been written by “Shekispeore”. We think that the quality of the two plays—on the one hand, the spontaneous, direct, vibrant verse of Khamlet, and on the other, the flacid, ponderous, convoluted meanderings of “Amlet”—speak for themselves. Those who persist in being Doubting Thomazeds would do well to consult the Central Federation Mediaeval Archival Database on the meagre, unconvincing amount of information extant on the existence of this Shekispeore, and compare it to the testimonials of the Declassified Approved-For-Aliens pre-Khitomer Personnel Rolls on Wil’yam Shex’pir.

It remains a fact, though, that these forgeries were as thorough as they were malicious: gigabytes of allegedly Industrial Age back-dated so-called Shekispeorian Criticism were fabricated, and the works disseminated as part of a well-organised campaign. This campaign appears to have succeeded far beyond its initiators’ anticipations. For better or for worse, works like Amlet, for all their crudity, have acquired a certain resonance amongst citizens of the Federation, and Terrans in particular. This is no doubt due to their pseudo-mediaevalist parochial appeal, which has rendered these incisive masterpieces of sociopolitical analysis into innocuous picturesque period pieces—a genre favoured on Terra (and Betazed) much more than on planets like Vulcan and the Human colonies.

And of course, the notion of a mass-scale falsification of historical records did not originate with me in 1996. There are Quora users right now, claiming that the bulk of our accounts of Ancient history passed down in mediaeval manuscripts are Carolingian falsifications.