Kakashi didn't kill Rin on purpose. Here's a reasons why.
Remember when Kakashi showed Zabuza the Chidori with the intent on killing him but Haku took the blow?
Zabuza &' Haku, Both Hidden Mist Ninja.
Zetsu said Rin &' Kakashi were under attack by hidden mist ninja
Remember when Zabuza trapped Kakashi in the mist jutsu?.
In Naruto Volume 3 #26, Kakashi says, "Blast it... I've been so worried about Naruto &' Sasuke, I hadn't considered...how long it's been since I had to fight under conditions this adverse!"
Maybe Rin &' Kakashi were under the same mist justu by the mist ninjas &' just like how Kakashi went for the kill on Zabuza &' Haku got in the way, Maybe Kakashi was set in on another target but Rin was mistaken for the target due to the mist?
Obito clearly said "Because you let Rin die" With that, we could assume that Rin wasn't in "her right mind"(he failed to protect her) or she was under some genjutsu, or that the Kakashi we see striking Rin wasn't really Kakashi.?