In the Star Wars universe, redemption for the villains usually goes hand-in-hand with another thing: death. In The Rise of Skywalker, that pattern holds true, with Kylo Ren finally finding his way back to the Light, but meeting his doom soon after. So how does Kylo Ren die? That's actually a more complicated question than it first might seem.
Officially, Kylo Ren makes it all the way to the end of the movie, when he joins Rey to try to defeat the resurrected Emperor Palpatine. Their Force bond is co-opted by Palpatine, who sucks all the power from it to revive himself and his powers fully before tossing an unconscious Kylo Ren aside into a pit in the ship. Rey is forced to defeat Palpatine on her own, using Luke and Leia's lightsabers to turn his own Force lightning on him, but the energy required costs her her life. Kylo Ren is able to climb out of the pit, finds her body, and, in a reversal of an earlier scene in which she heals him, revives her with the Force — which almost immediately costs him his own life.
Depending on how you interpret the dichotomy between Kylo Ren and Ben Solo, however, it's arguable that "Kylo Ren" dies long before this. The idea that a fallen Jedi's Dark identity "kills" their heroic side is one that dates back to the original trilogy, when Obi-Wan Kenobi describes Vader's fall as Anakin Skywalker being killed by the rise of Vader. From that interpretation, Kylo Ren dies earlier in the movie, when he hears his mother's voice calling to him through the Force as she dies and when he sees a memory or Force projection of his father reassuring him that Ben Solo is still alive. Either way you interpret it, Kylo Ren definitively dies in the movie, as does Ben Solo. In a way, it's a fitting end: much of his arc was tied up in trying to imitate his grandfather, and in the end, he did, but by embracing Anakin's legacy, not Darth Vader's.