How did this begin? Did it really start with Bugs Bunny?
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).
What's up is used in "The Sea-Wolf", written by Jack London and published in 1904.
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:
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: