The basis for Elvis being called a racist is that he allegedly said:
“The only thing niggers can do for me is buy my records and shine my shoes.”
This was supposedly said in June of 1957, while in Boston or during an interview with Edward R. Murrow on CBS.
But according to various sources Elvis had never been to Boston at that time and apparently he never appeared on the CBS show, you'd think there would be footage of it floating around if he had.
Here is a historian debunking the alledged statement as an urban myth.