The insult is probably older than English.

Cross-linguistically speaking, if something is seen as bad, you’re bound to see people using it against things and people they dislike. Doubly so with the imperative or the volitive, as if expressing a wish; like, “may bad things happen to you”.

So odds are “fuck you” and similar expressions (like, say, “fuck thee") are as old as the verb “to fuck” itself. And the verb has cognates all across the other Germanic languages; see German “ficken” to fuck¹, Swedish “focka” *to copulate, Icelandic “fokka” *to mess around, to rush². This shows the word is most likely present already in Proto-Germanic (500 BCE ~ 200 CE), so it predates English. (It’s usually reconstructed as *fukkōną “to copulate, to assail”.)

And odds are it was already vulgar for a long, long time. That’s because it is barely attested in older times; even being such an old word, it’s first attested in English in 1475, almost yesterday. This mismatch between being an old word vs. barely recorded shows it was something people would say but not write down³.

1. Perhaps not surprising you can use “fick dich” fuck you in German pretty much like you’d use it in English.
2. Icelandic “fokka” likely went through a semantic shift like “to copulate” → “to copulate promiscuously” → “to do things haphazardly” → “to mess around”, “to rush”. It’s similar in spirit to the English expression “to fuck around”.
3. You see the exact same mismatch with Latin “merda” shit. We know it’s inherited from Proto-Indo-European *(s)merdh₂ “stench”, and Romance speakers still use it all the time; and yet you barely see it being written down in Classical times, I think it’s only attested from Martial (whose whole shtick was to be offensive).