Kakashi did not intentionally kill Rin; she sacrificed herself by intentionally jumping in front of Kakashi's Chidori (Lightning Blade) to protect the Hidden Leaf Village. Rin had been forcibly made the Jinchuriki of the Three-Tails by the Hidden Mist Village's shinobi, who planned to unleash the Three-Tails' power inside Konoha (the Hidden Leaf Village), causing massive destruction. To prevent this, Rin chose to die by Kakashi's attack, essentially committing suicide to save the village. This tragic event deeply affected Kakashi and led to Obito Uchiha becoming vengeful as he misunderstood the situation, thinking Kakashi had betrayed them by killing Rin intentionally. Rin's death was a sacrifice to protect her village and friends, not a result of Kakashi's betrayal or cold-heartedness.
