Lee is a jerk?
Lee is actually one of the most influential characters in the plot because of his scheming. He pulled the strings behind the scenes in TK1-2, was absent in 3 but then did it again in 4,5 and 6.
Lee isn't just a somebody, he's one of the largest shareholders in G-Corporation, thus making him one of Zaibatsu's most threatening enemies because he's able to stand toe-to-toe with them.
Plus Lee has always been considered Kazuya's rival. He is everything Heihachi always wanted but failed to recognize. He is basically Heihachi, he plans things from behind the scenes, uses brains instead of brawns, use alliances to do the fighting for him but isn't afraid to go toe-to-toe with anyone. The only flaw in Lee is he isn't a Mishima by blood, not to mention, Japanese.
Why does Kazuya hate Lee? Here's a better question, what doesn't he hate?
Lee are one of the few characters who could properly control the Zaibatsu AND the G-Corporation beside Heihachi. Look at Jin, he got the Zaibatsu and what happens? World War. Too much power for a 21 year old.
How isn't Lee relevant to the plot? Well he's one of the few who could fight a Mishima and not go down easily (only Hwoarang and Paul have ever fought and come to a draw with a Mishima, Lee almost defeated Kazuya but lost in T4; Major improvement from T2).
Ok, I'm a Tekken nerd, but heck.
I got two fists.. Don't make me use my head as well!