I know how strong I am and how fast I can run. I have a rough measure of my intelligence and also how persuasive or good looking I am. I know that I can or cannot cast spells and what skills I possess. I have a name. What do I know about my character?
Not much. A lot of what we're doing when we build characters is we build them from the inside out. We fill our character's attributes and traits, but we have no clue what culture our character comes from (and no elf or dwarf is not a culture), and because we don't know really what culture the character comes from, we don't have any clue what the character does or does not believe.
Before you start hunting out the systems that do deal with culture, another gentle reminder that I am looking at general trends and I am looking at systemless approaches to the issues I account for. While it would be nice to just say "everyone play game XX", I don't think it's practical. To be honest, I don't really want to prescribe a one system-fits-all solution for any of this. Systems are great, but what I care most about is how I'm thinking. A lot of things get done because that's how they've always been done and very little reason beyond that. I want to look at these assumptions and pull the thread a bit.
Not having a cultural reference point is rarely a problem in modern or near future games because we get to use the time we live in as a reference point. But the underpinnings of even a stock fantasy world are such that lacking culture is the equivalent of lacking character. We can try to overwhelm this lack with a big backstory, but I've a rather dim view of doing a bunch of story before we even start playing. After all, we're sort of playing to tell the story, right? If you come into the game with 10 pages of background, you have a lot of detail, but all that history in practice can really stifle play. Plus, in order to really bring it into the story, the whole table really needs to know it, so that they may steer play to interact with it. Multiply the big back-story by several players, and you soon have background that makes actually playing a bit difficult.
Without this, characters are hard to get a grip on until we actually play for several sessions. We are forced to make decisions, and merciful GMs usually let people undo some of their choices early in a character's life to make sure a player is playing the character she wants to play. That's a fine workaround to characters lacking background, but fundamentally I think the problem is that we build characters as individuals first. It is very common in our fantasy gaming to build an imaginary person and then bolt culture (if we have any reference points for it) on later. That makes sense, right? Our stories are those of individuals making some change in the world, so what is wrong with building that component?
The problem with that is that no one exists alone. Try as you might, you exist, right now, in a political, geographical, and social context. Humans are first and foremost creatures of culture. To understand any human fully, you must understand the place and times in which they live first. Knowing this will let you know where they are typical, where they are atypical, where they struggle and where they excel. Every person is a product of their society. Even if you play the typical anti-social loner character, that character's anti-social behavior is reflecting off of some other culture. Your anti-social loner who has left a culture of nomadic desert people likely carries a different set of beliefs and traits than my anti-social loner who comes from a agricultural culture in temperate climates.
Culture informs the basics of what we believe and how we perceive the world. Culture implies a history, as culture is often a reaction to events happening to a people. Culture also implies culture, as significant shifts in context (racial, geographical, religious, et al.) create alternate histories (really alternate views of history) which in turn create different reactions which then become different subcultures.
In this sea of culture and history are the individuals who live in it right this moment. Your character is one of them! Now that you know you came from the nomadic desert people, what can you say about your character? How does your race affect that context? Did folks treat you differently? Why have you left it? Do you identify with it or reject it? When we start our character creation culture first, all of these questions emerge quite naturally. Once we get a sense of where a character is from, determining the character's attributes now take a whole new dimension. Determining who are individual is in the context of other people in a time and place makes everything else make sense. The great thing is that it even gives us purchase when we reject things! Rejection and acceptance of cultural norms are both equally powerful in terms of character definition, and the act of simply accepting or rejecting everything from our culture is an easy way to make a powerful character. It might seem unsubtle, but there's even room for nuance by approaching matters in detail.
The problem here is that using culture to start character creation requires having cultural hooks to begin with. There is a lot of setting detail that fantasy games have, but for the purposes of actually playing games, having to read all that material isn't going to work for most people. "You need to make a character, so go read these forty pages first" is not very appealing. How then, do we approach culture-first characters?
I've created some Gameable Culture tools that will get you started. Condense the relevant cultures of your setting with these tools, and make a 1-2 page reference for your players. They will not have the entire history of your setting, but they'll have enough to ground them and start playing. The trick here is that you don't need a setting's entire narrative to engage with it. Evocative hooks work incredibly well here.
It's important that you portray each culture as worthy to itself. If there is a culture where lying is acceptable and corruption is commonplace, don't write their defining statements as if they think of themselves as liars — find the reason that they find lying acceptable. Everyone is the hero of their own story, so make each culture worthy to itself without judging from the outside.
The upside to this approach is that you then create room for characters from differing cultures to productively disagree. If my culture sees yours as liars, we might be able to talk about that thing. Your character might choose to explain how you see it, and I can accept or deny it. The key isn't that there are right or wrong answers, it's that we create interesting, layered hooks for play.
I know that it seems I've delved into moral relativism, but let's put this in the proper context: We are playing games that produce stories. What makes stories interesting are detail and conflict. By providing the hooks and details for cultural interaction, what we are doing is creating a space for our game to have richer detail and conflict. I don't believe that we need to believe the things our characters believe. I think sometimes the most interesting characters to play are those who see the world very differently. But to run those characters, we need the groundwork for building how they see. In all cases, the way to know how someone sees the world is understanding what it is that they've been shown, and what they have already seen. Understanding their cultural context is the most sure way to get that.