hmm...well, he doesn't exactly kill L himself. He USED people, so that in the end, Rem was forced
to kill L (because of Rem's love for Misa).
At least I think that's how it happened...nee? I just watched the live action movie, and it was waaaaay different.
(differences including L faking his death, coming back, captching Light trying to kill his dad with the DN, Light dying by way of Ryuk, but then L dying in 20 days...me and my aunt who watched it with me are still confused as to why Rem made L die in 20 days and not instantly, lol. I dun get it...