Irrational logic? That's an oxymoron.
Sasuke wants to kill Naruto for three central reasons:
1. Naruto is the embodiment of what he feels the need to erase. The ideals of the past and the tailed beast, as well as being the only one who can match or exceed him in power and is capable of stopping him.
2. He believes that the Hokage is a conduit, a lighting rod, that draws in all of the world's darkness and embodies it so it doesn't affect the rest of the world. The first step in doing so is submerging yourself in the darkness to the furthest possible point. Itachi, in Sasuke's eyes, came the closest to doing so and was a true Hokage, but made a mistake: He couldn't sever his closest bond and kill Sasuke, so he failed to permanently change the world for the better. Now, Sasuke won't make that mistake, and sever his closest bond, I.e. killing Naruto, allowing himself to fully become the lightning rod of darkness and truly changing the world.
3. Being that the true Hokage must shoulder all of that darkness and hate himself, it's burdensome. It's not an easy thing to do and effectively ruin who you are. In a way, he may see himself as doing a favor to Naruto. By killing him, which is the only way headstrong never-give-up Naruto won't become Hokage, he saves him from having to make that sacrifice.
The third one is just my interpretation.
