Read more about Kazuya Mishima at: Wikipedia
Official Site: Namco
According to Namco's official backstory, Kazuya Mishima is the son of Heihachi Mishima, the long-time Chief Executive Officer of the Mishima Zaibatsu (a powerful worldwide conglomerate), and grandson of Jinpachi Mishima. He has a foster brother, Lee Chaolan, who was adopted by Heihachi Mishima, and also a half-brother, Lars Alexandersson, Heihachi's illegitimate son. Most importantly, he is the father of Jin Kazama, through a relationship with Jun Kazama.
When Kazuya was ten years old, he was thrown off a cliff by his father, who claimed that his son was weak. If Kazuya was to be his father's successor to the Mishima Zaibatsu throne, he was to not only survive the fall, but climb back up the rocky cliff as well. Kazuya nearly died from the ordeal, but his survival was assured through a deal with the Devil, who promised him enough power to take his revenge on his father.
In the years prior to the events of the original Tekken, Kazuya entered martial arts tournaments all over the world, becoming an undefeated champion (only Paul Phoenix ever managed a draw with him). When the first King of Iron Fist Tournament was announced, Kazuya took the opportunity for vengeance. It is during the tournament that the original game takes place.
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