Kimahri's a hard fella to know. As a big, buff, blue Ronso, he's pretty darn intimidating, and since he rarely speaks, it's hard to think of him otherwise. But if you look at how protective of Yuna he is, or his few kind words to Rikku, or the promise he kept for Auron, it's easy to see that Kimahri's really just an old softie.
I wish I had more to say about Kimahri, but really that's about it. ^^;; His only other really significant character trait is his honor, which makes him a bit of an outcast among his Ronso tribe. The other Ronsos like to tease Kimahri mercilessly for being "hornless," and Kimahri is deadly and focused about regaining his honor in that regard. I would bet his loyalty and warrior's spirit are why Auron admires and confides in him.
"Kimahri think Rikku should stay Rikku."
Do Kimahri and Auron ever actually exchange words? I can't remember a scene, but I still think there's a relatively interesting relationship there. After all, Kimahri is the one who fulfilled Auron's promise to Braska, taking Yuna to Besaid when Auron's wound from Yunalesca prevented him from keeping that promise himself.
We never really know whether Auron knew Kimahri before that, or whether Kimarhi was a friend of Braska's. Did he just happen to pass by and see Auron dying in the snow, and do him this favor? Or had they been friends before? Either way he did more than just take Yuna to Besaid; he looked after her from day one. You could say he was her first guardian, before she ever knew she needed one. Kimahri watched over her as Auron watched over Tidus in Zanarkand. Since they essentially performed the same task, Auron and Kimarhi may have realized they have a lot in common, if only in warrior's pride.
Is that why Kimahri is the only one to really get a good-bye from Auron before Yuna performs his sending? It makes me think there may be a little more to their relationship than just the fact that they both looked after Yuna. Maybe they were friends. Maybe they were just aquaintances. In any event, they seem to have a lot of respect for each other, even if a word is hardly ever spoken.