The story of the famous Lee brothers, known in their hometown as the Twin Dragons, is probably quite exciting. It's too bad we know very little about it thanks to Capcom not ever giving Street Fighter III much backstory. Yun Lee, known as the Storming White Dragon, is the more outgoing and obvious of the twin brothers.
Yun is related, in some fashion, to two famous martial artists. He was raised by his "relative", a man known only as Lee who had participated in the first Street Fighter tournament. At some point after that he came under the care of the world-famous assassin Gen. Gen had retired from his killing ways by then and he was running a restaurant known as the Genhanten. Unfortunately, Gen had developed leukemia. His plan was to go into the world and find a perfect deathmatch to end his life in glorious battle. Whether or not he succeeded or he died due to the leukemia is unknown in Street Fighter canon.
What is known is that Gen spent some time tutoring the twins in his fighting style. He then passed on the management of his restaurant to the teenage boys. Sometime after this the two began to take it upon themselves to guard their town (really a section of Hong Kong) from criminal mafias and corrupt politicians.
Yun has a sort of devil-may-care attitude and a huge ego. He likes to show off, playing up his fighting skills and generally making a spectacle of himself. It is also highly possible he is the better martial artist of the brothers; at the very least, this is the popular opinion. Yun seems primarly interested in protecting his own 'town' above other concerns. He has a girlfriend (Hohmei) but he seems to treat her more as a distraction and nuisance then with deep affection.
Yun is a consummate martial artist. While not on the top tier he is nonethless very powerful, having been taught several moves by one of the most powerful martial artists in the world. Yun is constantly improving himself, though the extent of his potential remains unexplored. He relies primarily on high speed moves and mobility. He is also an expert skateboarder, but prefers to fight with more traditional kung fu styles.