I believe people's traits are not particularly inherent to their social class, race, gender, nation, or sexuality, but what those axises do change is people's relationships with those traits, and what society in general expects of them.
So here is a process.
1. Pick the traits you want for your character. I don't mean black, straight, and moneyed, I mean like they're recalcitrant, competitive, intensely focussed on personal hygiene, and they compete in dog shows. It doesn't have to be anything wild or "different", just however you would normally make a character. Don't think about how to write a woman, or an Asian American, or a post-operative transexual. Just think about who they are as an individual right now.
2. Determine how fixated they are on the political axis of their identity and what they think they're "supposed to be" based on that (note, this may not be the standard, this is their personal thing). You'll hear people say things like "people like me don't go to college" or "that's fine for them, but I don't think it's appropriate for someone like me". Does your male character shop alone for fabric because he loves sewing, but feels like it might make him a little less of a man? Does your black character feel guilty about liking Tarantino more than Spike Lee? Does your lesbian feel a little vain about how many men's head she turns, even though she has no interest in them? Is your quiet domestic female proud of herself for doing so well at being the perfect wife?
Conflicts and double standards of identity are great for character development.
3. Once you've determined how they feel about the places they deviate or conform with their idea of their axis, think about the social pressures this axis brings on them and how they react to that. Say you have a very aggressive woman who loves to fight and is pretty proud of how well she does- what does she do when people whisper behind her back, or when her friends ask her why she just won't settle down and be normal? How does your shy young black man who loves to compose waltzes react to his classmates and his relatives trying to engage him in verbal sparring, or get him to play music that they think of as less white? Does your character hide and try to blend? Do they dig in their heels in resentment and be what they are more visibly? Do they lose confidence in themselves because deep down they feel everyone around them is right?
Conversely, how does that shy young black man feel when a white professor congratulates him on not confining himself to "the ghetto of black music"?
Society is no more uniform than people are, but statistics do come into play. If your character deviates, they will eventually catch flack from one side or another (or from many sides. You can't please everyone, but sometimes it feels like you CAN displease them all). If your character conforms but feels ambivalent, they're eventually going to be rewarded for something they're not sure they feel good about. If they naturally conform, they may internalize the expectations so thoroughly they have trouble believing anyone COULD deviate ("oh that's just silly! All women are inherently nurturing!") and get flustered when they run across it.
Not only do these scenarios help you round out a character, they make for great scenes.
So, that's my thought process on it, and I hope it's something I do right by. I also highly recommend Nisi Williams' "Transracial Writing for the Sincere" for people who want to be more multicultural with their characters, but are scared of doing it wrong.