There are a bunch of stories about time-travelers going back to find out who wrote Shakespeare's plays, such as Henry Kuttner's "The Comedy of Eras" and Nelson S. Bond's "Much Ado About Pending"; but the one you described is "The Muse" by Anthony Burgess of Clockwork Orange fame.
This was a short story in which a time traveller -- might have been an English Lit professor
"You're quite sure," asked Swenson for the hundredth time, "you want to go through with this?" His hands ranged over the five manuals of the instrument console and, in cross-rhythm, his feet danced on the pedals. He was a very old man, waxed over with the veneer of rejuvenation chemicals. Very wise, with a century of experience behind him, he yet looked much of an age with Paley, the twenty-five-year-old literary historian by his side.
-- travels back to Elizabethan England to determine once and for all who wrote the plays and sonnets of "William Shakespeare".
"We have to check up on history," said Paley, mumbling a little. His own quest seemed piddling: all this machinery, all this expertise in the service of a rather mean enquiry. "I have to know whether William Shakespeare really wrote those plays."
He lands in a society where everyone distrusts everyone else, where informers may rat you out to the royal secret police and the dominant sentiment is fear. Finally he does meet Shakespeare, but it's not a happy discovery. Apparently the famous playwright is a space alien... and not the friendly kind, either.
"'Tis not seemly to read a gentleman's private papers lacking his permission." Paley spun about to see, dancing in the air, a reproduction of the Droeshout portrait of Shakespeare, square in a frame, the lips moving but the eyes unanimated. He tried to call but could not. The talking woodcut advanced on him—"Rude, mannerless, or art thou some Privy Council spy?"—and then the straight sides of the frame bulged and bulged, the woodcut features dissolved, and a circle of black lines and spaces tried to grow into a solid body. Paley could do nothing; his paralysis would not even permit him to shut his eyes. The solid body became an animal shape, indescribably gross and ugly—some spiked sea urchin, very large, nodding and smiling with horrible intelligence. Paley forced it into becoming a more nearly human shape. His heart sank in depression totally untinged by fear to see standing before him a fictional character called "William Shakespeare," an actor acting the part. Why could he not get in touch with the Ding an sich, the Kantian noumenon? But that was the trouble—the thing-in-itself was changed by the observer into whatever phenomenon the categories of time-space-sense imposed. He took courage and said:
"What plays have you writ to date?"