Classic music fans can breathe easy on this one. Although some worrying criticisms against Elvis might hold some merit, racism probably isn't one of them. Most frequently, people bring up a quote by Elvis from the '50s, something along the lines of "The only thing Negroes can do for me is shine my shoes and buy my records."
We say "something along the lines of" because nobody knows exactly where the quote came from or how it was worded -- most likely because it was completely made up by someone spewing bullshit. The furthest anyone can trace it back is to a 1957 article in Sepia magazine, which apparently got it from an interview with an "anonymous person on the street."
Elvis himself vehemently denied ever having said anything even remotely like that when he was confronted about it, and his many black friends backed him up on it. And we know what you're thinking -- "Oh yes, the black friends defense, very original." But honestly, Elvis had many, many black friends. So many that he attended a black church headed by celebrated civil rights activist Rev. W. Herbert Brewster. If the notorious quote was accurate, that philosophy doesn't seem to have made its way into Elvis's everyday life at all.