Why Did Chancellor Gorkon Say...

No, he’s implying that the Klingon language allows for nuances which are not present in the English language, and as such, when Shakespeare’s works are translated to Klingon, they have more depth, more room to breath, thus allowing for a more enjoyable reading experience.

because of WS greatness, he is "claiming him" for the klingon race

he knows it is not really true

Well Spock implied he was a direct descendant of Sherlock Holmes so why not? :-)

I expect that the real reason was a reference to how Nikita Khrushchev claimed to Vice President Richard Nixon that Russians did everything better than Americans when they met.

I assume he's implying that it's originally Klingon. But is that ridiculous? Yes and no...

I think it's a bit of a Cold War joke. Chekhov got some similar lines on the original show, about how such-and-such a thing was Russian when it wasn't. He claimed the Cheshire Cat was from Minsk, for example, and that scotch whisky was invented in Leningrad. Here's a video: https://www.youtube.com/watch?v=IC6W8J0j8Co&t=125s

This thread talks about it a bit more: USSR brainwashing was at play to convince their people that Russia was the greatest country in the world. https://scifi.stackexchange.com/questions/17707/does-chekov-believe-his-claims

Because of the Cold War parallels present throughout Star Trek, but particularly with The Undiscovered Country, this line about Shakespeare being Klingon is, in my opinion, meant to be an echo, extension, or reference to the USSR thing.

Of course, even without brainwashing, countries tend to develop an overinflated sense of self-worth. Ask people about which lineage is responsible for the telephone or heavier-than-air flight, for instance, and you'll get an interesting stew of answers.

In short, Gorkon probably was claiming Shakespeare for his people, and Star Trek VI was setting up a parallel to the real world in an amusing, if over-exaggerated, way.