In "Babylon Revisited," what are the fatal flaws in Charlie's character?

I would say that "Babylon Revisited" is, more than anything else, about how impossible it is to fully erase one's past mistakes. Indeed, it should be noted that many of the story's most important events and plot points happened in the past. This is, more than anything, about the lingering aftermath, which never fully resolves the way its main character wishes it would.
This story's basic plot follows Charlie as he seeks to regain legal custody over his daughter, Honoria. Over the course of the story, Fitzgerald slowly reveals more and more details about what brought him to this point in his life, and just what sort of demons lurk within his past.
Looking backward into the past, we'll find that Charlie possessed a number of qualities, all very much emblematic of the image associated with the Roaring Twenties. He was wealthy and ostentatious; he made money and he spent money with reckless abandon. He drank heavily and was an inveterate social carouser. This was a very turbulent kind of lifestyle, and that turbulence was reflected in his marriage with Helen. Returning back to the present, we see that his sister-in-law blames him for Helen's death, and it is this same woman he now needs to convince to return guardianship over his daughter.
So in answering your questions, you can discuss various character flaws individually, and they certainly apply. There's his drinking and his irresponsibility, but more than anything else, I think, it's the totality of that entire hedonistic lifestyle that has led him to ruin. It cost him his wife and his daughter, and no matter how much he might try to pick up the pieces and get his life back in working order, those stains are never wiped clean. He's still paying for them. We see this expressed symbolically with the intrusion of his old friends Lorraine and Duncan, which in the story's conclusion ruins his chance at regaining Honoria's custody.