Prior to all the events on the games, Lee Chaolan was just a street kid striving to survive the harsh reality, up until Heihachi noticed the toughness on that kid. As such, the leader of the Mishima Zaibatsu went on to raise the kid as his own. Although he actually befriended Kazuya for a brief time, Heihachi's harsh training regimen, as well as the contest for the leadership of the Zaibatsu, ended that bond between the two youths.
Because Lee was initially stated to be one of the contenders to heir of the Mishima Zaibatsu, that was probably the reason for his entry at the first King of the Iron Fist Tournament, in order to confront Kazuya.
After being soundly crushed by Kazuya in the first tournament, he was forced to serve Kazuya as an underling, to his utter chagrin. Lee was then approached by by chinese elder Wang Jinrei, who warned him about Kazuya and asked his help in returning honor to the Mishima Zaibatsu. Lee instead chooses to help his boss by delaying Heihachi long enough for Jun Kazama to defeat the Devil within him.
Eventually, Kazuya announces the second King of Iron Fist Tournament. Lee eventually meets a really angered Heihachi; he is then defeated and, after Heihachi claims back the Zaibatsu's control, is banished from the family.
ARCADE ENDING (non-canon): Lee wins the tourney and takes the Zaibatsu leadership, while Heihachi and Kazuya were put behind bars. As he imagines that sight, he makes a thumbs-down hand gesture, while proudly smiling.
Twenty years have passed after the second tournament; for quite some time, thoughts of revenge consumed Lee. But he realized that it was pointless to kill an old man, who probably had only a few years left in his life at best, so he decided to follow to a mansion in the Bahamas.
One day, he eventually discovered that the G Corporation has suffered significant damage from an attack by the Mishima Zaibatsu. He also came across information that a new lifeform under the Zaibatsu's development would be completed by next year, even though it was lacking an essential component...
Oddly enough, a fourth King of Iron Fist Tournament was announced, which triggered Lee's desire to fight, along with the disdain he had for the Mishima clan.
In order to prevent the Zaibatsu from discovering his true identity, Lee donned a new persona: Violet.
During the tournament, he met Kazuya, who was thought to be dead since the end of second tournament. Because of the shocking reveal, Lee failed to maintain focus and was defeated by Kazuya.
ARCADE ENDING (non-canon): Lee wins the tourney and reveals his true intentions: to advertize his invention, Combot. However, as he calls Combot to the arena in order for a demonstration, he falls down the stairs and his whole programming glitches out; he then charges against Lee, in combat position;
We then see a secretary entering Lee's division, with a document about the prototype for the new Combot version and the information that Combot has taken 10.23 seconds into experiment to unleash a 50 hit combination against his guinea pig: Heihachi Mishima. Pleased with what he hears, Lee says "Excellent".
After the tournament, Kazuya and Heihachi were at the Honmaru - who was then invaded by a heavy swarm of JACK-4 units. Heihachi is thought to be dead after one of the JACKs exploded. Lee was preparing himself to take over the Zaibatsu, but someone else had done that.
Thinking that 'someone' was his adopted brother, he decides to enter the fifth tournament in order to settle the score.
However, he later discovers that it was another person, Jinpachi Mishima, who led the Zaibatsu. Upon discovering this, he forfeits the tournament out of lack of interest, and goes back home.
ARCADE ENDING (non-canon): enjoying a nice massage at the pool of his mansion, he snaps his fingers to summon his new waiter: Heihachi. Lee then asks him 'Two Fingers', and as Heihachi walks away, he calls Lee a bastard.
Hearing that, Lee threatens him by pressing a button at a remote control. Many beeps come from Heihachi's bow tie, which makes the old fighter recall the explosion at the Honmaru in a panic. Heihachi runs away frantically, as Lee lays down again and says 'Excellent'.
Soon afterward, Jin Kazama takes control of the Mishima Zaibatsu, only to bring the world to chaos, with the G Corporation entering as an insurgent force against the Zaibatsu. Surprised by G Corporation's sudden millitary action, Chaolan, a major shareholder, conducted a private investigation, and discovered that Kazuya was the driving force behind G Corporation's actions.
After obtaining information that Kazuya himself would take part in this tournament, Chaolan, using a shareholder's preferential treatment ticket, enters the tournament in order to get close to Kazuya.
At some point, Lee becomes a somewhat important part during the Scenario Campaign mode. Acting at first as a boss, Chaolan becomes a partner to Lars and Alisa, since they have the same motive against the Mishima Zaibatsu and the G Corporation, giving him intel about the two organizations' other schemes.
ARCADE ENDING (non-canon): after winning the tournament, Lee is seen playing golf, while Heihachi, Kazuya and Jin are imprisoned in rockets. Lee putts a golf ball, that falls near the Mishimas. Heihachi taunts him, but as the ball explodes, Heihachi, along with his son and his grandson, start screaming for help, all while Lee putts even more golf balls. Eventually, the explosions trigger the rockets' wicks and the Mishimas are sent into the air, to their demise. As he watches the fireworks, he takes off his sunglasses and says "Excellent".