In hopes of not sounding like a pretentious prick, I'd like to write a little tutorial on how to write compelling characters to roleplay. I realize that some people are content with one-dimensional PC's that are a simple archetype, but I know there are those that want to go beyond being the stereotypical soldier, smooth talker, techie, et cetera. If you fit into the latter category, read on for some character building advice. If not, you can tell me to fuck off and we can go our separate ways.
The first thing that I think of is what I want my character to end up as, and start thinking about the early life needed to develop into what I want. Who they were has an enormous impact on who they will become. Furthermore, what works for me is actually to think of an archetype. I know I implied above that achetypes are bad, but this forms the basis of what your character is skilled in and what they are like physically. After choosing your archetype, you can begin developing your character; there is a great website here that lists 100 questions you should ask yourself about your character. There are seven categories that develop your character, beginning with the basics, like where/when they were born and what their occupation is, and continues on to ask what their youth was like, what has influenced them, what their beliefs and general opinions are, their relationships with others, their likes and dislikes, their self image and other miscellenous questions. Not all questions may apply to your character, but they are an excellent method to get you thinking about who your character is, not what they are.
This sentence is legitimately the only question that needs to be answered to create a compelling and rather unique character: Who do you want your character to be? Not what, but who. Maybe they're looking for wasteland renown, a mountain of bottle caps, or just a simple, steady and safe existence. Determining what it is they seek will help answer the question of where they went or where they are now to achieve those goal(s). What did they have to go through to get there? Was it everything they hoped it would be? How did they handle things not going to plan? This is something you must realize: these characters are not (or should not be) static, they are dynamic. It is OK for them to undergo change; in fact, it's strongly encouraged. Something happening to change their worldview (for better or for worse) is better than them having the same worldview their whole life. Perhaps you are roleplaying with them (and they are adventuring) to find a new worldview.
After you figure out your character's motivation(s), answer how they got to where they are now. This means both how they got to their current location, and how they became who they are. For example, maybe your character was a farmer that hated the dull life, so he up and left his farm in Arroyo, hopped aboard a caravan and made his way to New Reno. That answers how he got to his new home. Looking for some excitement, he went to the Desperado casino but racked up a large amount of debt. He went to the Mordinos and arranged to help them synthesize new chems using his knowledge in agricultural chemistry (which he knew from his life as a farmer). This answers how he became affiliated with his faction. Keep in mind that not all characters have to belong to a faction, but it can help solidify their ideals and personality.
These things establish your character's past, but it doesn't answer the question of what they're doing now. Maybe our farmer paid back his debt and left New Reno for a second shot at a better, safer life... or maybe he stayed to become their top chem synthesizer after Myron's death because of the lucrative benefits that come with the job. If you answer the question of what they are doing now, 1) it can determine what kind of person they are (homebody or adventurer, someone who incites or avoids conflict, et cetera) and 2) it will be easier to determine what they will be doing in the future.
To sum up, figure out: 1) the "basics" of any person, 2) who they want be, 3) how they got to where they are now (location and affiliation) and 4) what they're doing now. If nothing else, keep asking questions to fill in the holes in the past, present and future. If you have questions, a) answer them for yourself or b) ask someone else what they think a proper path is; if it makes sense and you like it, go with it. If not, alter it so you do like it. Or if you have questions about the process, you can ask me below in the comments.