Why does Chancellor Gorkon think Shakespeare was Klingon?

It was a Klingon propaganda device.

The Klingons apparently appreciated the works of Shakespeare so much that they 'retconned' him to be one of their own poets rather than a human from Earth.

From Wikipedia's article on The Klingon Hamlet, sourced to the DVD commentary on the film Star Trek VI: the Undiscovered Country:

The film's director Nicholas Meyer said the idea for having the Klingons claim Shakespeare as their own was based on Nazi Germany's attempt to claim the Bard as German before World War II. A similar scene appears in the wartime British film "Pimpernel" Smith (1941) in which a German general quotes Shakespeare, saying “'To be or not to be', as our great German poet said." The idea had also already been used by Vladimir Nabokov in his novel Pnin, the eponymous hero of which taught his American college class that Shakespeare was much more moving "in the original Russian."

This same trope - people claiming certain things originated in the speaker's own culture rather than wherever they truly came from - appears a LOT in Star Trek as a whole. In TOS, Pavel Chekov often claims that various sayings, and even the Garden of Eden, originated in Russia, and Spock attributes quotes from Sherlock Holmes and Richard Nixon quotes to ancient Vulcans. Quark claims that "discretion is the better part of valor" is a Ferengi proverb, and Khan that "revenge is a dish best served cold" is a Klingon saying.

_{Thanks to ilinamorato and A.D for their answers to a similar question on another SE, which I used in writing this answer.}