I'm a little confused on the plot myself. I know little bits and pieces so i'll try and explain it the best i can..
.. from what i know Heihachi Mishima, was the one who held the original King of Iron Fist tournament(tekken 1). The winner would be awarded the Mishima Zaibatsu corporation as the prize.
At the end of the first tournament Heihachi was defeted by Kazuya. Then Heihachi came back and defeated Kazuya in the second tournament, allowing Heihachi to regain control of the Mishima Corporation once again. Heihachi, also discovered the devil gene hidden in Kazuya during the second tournament. He tried to gain control over it but failed.
So then after Jun Kazama(jins' mother) disappeared during the battle with Ogre.. Jin went to Heihachi (his only relative). Heihachi then heard about ogre and tried to lure Ogre out using Jin. He also figured that Jin's devil gene would be awakened if he were to encounter Ogre in battle. Heihachi, was basically tring to capture either of the two or both, but failed yet again. (as shown at the end of tekken 3).
I think Jin's hatred for the Mishimas has to the with the fact of being caught in between the rivalry of both Kazuya and Heihachi. This is why he learns a new fighting style, rather than using the style that his father and grandfather have mastered.