The Elite is generally born into privilege, not necessarily wealth but usually from some form of family that is highly respected for whatever it is they do. In a "Business" story, the Elite will come from a family of well off businessmen, in a fighting story the family will have some legendary warrior in its blood lines. The Elite has been raised their entire life to be the best in this category, perfection was not a goal that was sought after, it was what was expected of the character regularly. Not only that, but an Elite generally doesn't allow others to see just how hard they are working at achieving their goals, whatever they may be.
The key defining characteristic of the Elite generally speaking, aside from the family connection, is talent. They are a highly talented individual, and things will appear to come very easily to them. They don't need to work as hard as other people to get that key grasp on some new thing presented to them, their talent makes it easier for them to pick up. As I said before though, often times this "ease" is masked by the fact that they are just as hard working as any other character, they simply don't let it show, training in secret to give the visage that they are Elite. It often will come as a shock, in stories, for the other characters when they realize just how much work the character is secretly putting in, all the while trying to maintain appearances for family or personal pride.
Story wise, the Elite's story is usually one of self discovery just as much as any main characters is. They find a reason worth fighting for, and however misguided it is go all out for it. Their place as a rival usually means that they end up doing some sort of face-heel or heel-face turn during the story (sometimes both!) before helping the hero accomplish their real mission. Other common story elements can include things such as a massive inferiority complex, as as good as they are it never seems to be enough for their family, or the image they need to provide. What they do doesn't receive the same praise as the actions of other people, because it is simply "as expected" of someone with their elite status.
In an RPG...
In an RPG an Elite type can be a lot of fun to play, though they are somewhat hard to do mechanically. The trick to mechanically stating one out is to specialize yourself. You can't be both Elite AND cover a lot of bases. Not without a lot of extra XP from the GM. However, with the same points as everyone else, you can pull off an elite in a specific area simply by focusing your development. You are in essence Min-Maxing your character here though, so be sure to talk to your GM about this before hand. Min-Maxing for character reasons, with character plot, details, and challenges you want to come after you for it is one thing. Min-Maxing so you can "rock hard" in combat is another thing entirely.
Role Play wise, being an Elite is just as easy as any other character. It may take some work, but adopt a slightly arrogant air. After all, you've been groomed from greatness and come from a respected line. Detail things to say they look easy, refined, well practiced. Don't let others see just how hard you are working. If your Elite is a fighter, than when in combat describe what they do and give it that flair. Make it visceral so it will stick out in people's heads.
Most importantly, the thing you need to remember when using an elite type? Go big or go home. If you are doing everything you can to be as crazy effective as possible, you just aren't going to stick out. It doesn't matter if you fail 3 out of 5 times, or even 4 out of 5 times. People are going to remember that time you went for broke and won. The time you managed to pull off 10+ raises for damage. Then when you keep doing it, it will be remembered as something you just do. Keep in mind how competent you want your character to appear, and then gamble for it. Go after the big and more dangerous people.
In game terms, your Elite may die quickly. But if they survive, they'll become in a lot of ways the talk of the game. If they die, it sucks but you can move on. When they live? They become the things of legend. You just have to play into it, and play it up, while pretending to not be.