Because Hinata is Neji's Cousin
-and at the perimilary round at the chunin exams in naruto shonen jump, neji almost did. he went sort of easy on her but when hinata lost she had all these holes in her arms and she got knocked out and it looked like she died and she almost did.
but after neji lost to naruto in the final rounds neji's uncle (hinata's dad and head of the clan) talks to him and neji trusts his uncle and hinata and the hygua clan gets closer. and neji trained with his uncle for a while until he became a jonin and he becomes friends with hinata.
~.~.~.~.~.~.~.~.~.~.~.~.~
he wanted to but all the senseis jumpedso he wont kill her u can find a picture on Google
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Naruto and Neji fought in the third part of the chuunin exams simply because their names had been chosen at random. However, Naruto also had a personal motivation to battle it out with Neji, since he had witnessed Neji's fight with Hinata in the preliminaries and was furious at the brutality with which he fought and mocked her.
Because Naruto is the closest friend Sasuke knows and the only way to get the sacred sharingan is to kill your closest friend , there for Naruto and Sasuke rivals even though Naruto doesn't even know why.