Wizards and witches knew that Tom Riddle became Voldemort, and knew that Riddle was a Muggle name. Voldemort was ashamed, not only of his Muggle father, but that his mother had placed a spell on his father, who was never actually in love with her, and had married and seduced him. He wanted to be rid of this surname and wanted in invoke fear in people, thus changed his name to the alias Voldemort, but people still knew where he'd come from.
He said 'There's no such thing as good and evil, only power, and those too afraid to use it'. People might have known his heritage, but as someone who was so at ease with murder and domination, he had no hesitations throwing himself in the limelight and gathering followers, because people were either attracted to the power, or too afraid to say no, on fear of death.
So really, he was a hypocrite, but it didn't matter, because he was rarely disobeyed. He made out as if it was all about pure bloods etc but really, the dude had issues and just wanted to dominate everyone else.