There are two main points on which his legacy stands, neither of which have to do with his political career. The first is the persistent rumor that his great-grandmother was black. There's no concrete evidence for or against this, and as of when Francis Russell wrote his biography (in the 1960s) the Harding family were dead set against looking into it.
The story Warren's father stuck to was that a neighbor got mad at them and started the rumor that they were mixed race. The rumor was spread widely in the Blooming Grove and Marion communities where Harding grew up and then lived, and when he was running for president someone got wind of it and publicized it nationally to hurt Harding, which is how the rumor made it into the history books.
The point is, we don't know, and really it doesn't matter. What is certain, and what is consequential, is that WGH experienced racism in a very real and consistent way. Kids shouted racial slurs at him and his siblings at school, as a successful Marion businessman WGH was still barred from many of the clubs and associations he wanted to join, like the Masons, and his wife's father disowned her because she had "married a n---."
[Once WGH was a former lieutenant governor being eyed for the US Senate his father-in-law was more like oh heeeeey we're cool.]
Who's to say the effect this had on his life? He was certainly very ambitious, in the face of constant ridicule. And during his short presidency he gave a speech in Birmingham, Alabama calling for an "end of prejudice," the first president to mention civil rights in the South. [The speech was greeted with silence from the audience in front of him, and cheers from the black segregated farther away.] One could take the Tolstoyan view that it was simply time for the presidency to be for Civil Rights, but one almost must think that his lifelong experience of prejudice informed the speech.
The other thing Harding is chiefly know for is his flair for adultery. He married Florence Kling when he was 25 and she was 30, already divorced and with one son. Her father was a mean dude, and most people think she married the young, rising businessman as a way to secure a future away from him. For Warren's part, marrying Amos Kling's daughter might have given him some of the legitimacy and stability he needed to be a leading citizen in the community. It's doubtful they were ever madly in love with each other, but it seems they both knew what they were doing, and what they wanted, and ended up being a "successful" political couple, as these things go. She took over the business side of his newspaper, which immediately got out of debt, and was thought to be something of a political manager for him throughout his life.
Francis Russell, who wrote the biography I'm reading, is either a horrible misogynist or just hates Florence (or both!), because he never views their marriage with any nuance. He writes as if Florence trapped him into marriage against his will and then he was miserable forever. His reasoning: she was ugly, so duh. Here's something he wrote about their marriage about 10 years in:
"There were times enough when Harding wished that his wife were dead. Yet though he played with the idea, he could never assert himself to the point of leaving her. She was in her grim way part of him, a part he could not discard. Their dark and cluttered house represented home, with all the emotional overtones the word had for him. She, thick-ankled and withered, was no longer a sexual object, yet her illnesses distressed him. He had long been used to satisfying his physical needs elsewhere. She knew it, or at least sensed it, and was still woman enough to be torn with jealousy."
Yeah, Francis Russell can eat a bowl of butts.
But Harding did have numerous affairs, and visit numerous brothels. Most notably, he carried on a 15-year affair with Carrie Phillips, the wife of his friend and fellow Marion businessman James Phillips. The Phillipses and the Hardings were friends, and the two couples hung out a lot and went on vacation together sometimes (ewwwwww). Russell was writing his biography fairly soon after Carrie died (about 40 years after Harding) and got his hands on all the love letters from WGH she had kept. He quoted them extensively in the book, obviously, and it was set for publication when Harding's nephew filed for an injunction. The judge upheld it, but it was too late to re-write the book without the quotes, so they were simply replaced with dashes. The result is pages such as these:
It's a little funny because it seems just as damning as using the actual quotes would have been.
Anyway, the judge granted Harding's nephew a copyright to the letters, who then donated them to the Library of Congress under the condition that they would be sealed for 50 years.
Sooooo, fast forward 50 years. Harding's nephew has died and the internet has happened, and you can now read scans of the entire collection of letters on the Library of Congress's website. (Smithsonian Magazine has an article about the highlights here.) Jim Robenalt wrote a book based on the letters, The Harding Affair, and in an interview said that "ironically letters of adultery may be his salvation because they force the reader to take another look—a full look at Harding the man and Harding the statesman."
I'm not sure whether I'm getting a good look at Harding, either through Francis Russell's sexist/racist romp of a biography, or through further reading I've done on WGH, Florence, and Carrie. He seems to be an eternally unlucky man.