How did this begin? Did it really start with Bugs Bunny?
The Oxford English Dictionary says this colloquial use of the adverb up is very frequent from circa 1850, long before Bugs Bunny's 1940 debut. Two citations:
1838 E. C. Gaskell Let. 19 Aug. (1966) 37, I did not mention a word to Lucy but she must have guessed something was ‘up’.
1851 H. Mayhew London Labour I. 19/1 A shout in answer from the other asks ‘What's up?’
I found an antedating from 1813 in The history and adventures of Godfrey Ranger by David William Paynter:
And an 1819 in Right about face; or, Ben the Gordon boy by Emily Brodie:
As reported by Wiktionary, what's up has been made popular by the cartoon character Bugs Bunny, who used it as part of his catch phrase "What’s up Doc?" (around 1940).
Whatsup was used in the short story "The adventures of Shamrock Jolnes" from O' Henry (September 11, 1862 - June 5, 1910).
"Good morning, Whatsup," he said, without turning his head. "I'm glad to notice that you've had your house fitted up with electric lights at last."
What's up is used in "The Sea-Wolf", written by Jack London and published in 1904.
"What’s up?" I asked Wolf Larsen, unable longer to keep my curiosity in check.
"Never mind what’s up," he answered gruffly. "You won’t be a thousand years in finding out, and in the meantime just pray for plenty of wind."