I'm not sure if this is what teh OP had in mind, but I see 'bad' white characters as anyone where it seems like the Devs/Publishers said, "Let's make him a 20-35 year old average white guy because that's our key demographic and I don't know if they'd be happy playing as a black man".
Sam Fischer, Nathan Drake, Aiden Pearce, Marcus Fenix are all bad in this sense, even if they are good characters (well, Nathan is). Even Joel (who I think is great) is a bad example of using a white character. Their race is only used to maximise their demographic appeal.
The GTA5 guys are great white and/or male characters, because their race and gender are a key part of the story. It's about the perversion of the American dream and attitudes of masculinity.
I was one of those people annoyed by the continuing lack of a female GTA character, especially when we found out that GTA V would have 3 protagonists, all male. I was wrong though. The story totally justifies the use of the protagonists though I'd still like to see GTA 6 use a woman or a black guy who isn't a gangbanging hood-made-good.
Most historical games get a pass. John Marsden and Ezio are good because their racial background is an important part of who they are. The story whould have to be quite different if they were black, since they would face racism at every turn in the American wild-west or renaissance Italy.
I'd love to see a historical game with a story from a non-white perspective since it's a totally unexplored realm for stories. When we see the subaltern perspective being the subject of a game, we will show that gaming is getting serious about culture/art/storytelling.
Max Payne is double-plus-good, especially in 3. It's literally about the ongoing follies of a white guy and his unconscious privilege. He is a dumb white guy who gets in way over his head because he doesn't understand, and often has contempt for, the foreign culture he is in. He thinks he can save the world, but usually ends up fucking stuff up even more.
Far Cry 3 is double-plus-ungood. White guys rule and indigenous peoples are all clueless fucking peasants who need a spoilt rich kid to show them what's what. And no, getting killed while shagging voodoo priestess doesn't make it better, it makes it worse. Seriously, the indigenous people aren't even capable of fucking their own leader, because only your superior white-boy semen can provide the mesiah or whatever. And no it's not ironic. Anyone suggesting that it is should be forced to listen to Alanis Morissette on repeat for the rest of their natural lives.