Kuma wanted Afro to suffer for all his old friends that he got killed
Afro's master was going to fight Afro, but was interrupted, and Afro's master allowed Afro to kill him for the number two headband.
Kuma turns out to be Jinno, Afro's former friend who was thought to be dead.
Afro killed his master in order to become the #2 warrior.
Ninja Ninja: Damn, you're one stubborn samurai.
Kuma: Perhaps the rumors of the Afro Samurai are not true. That he cuts all who stand in his path. That his only movement is forward. But you just look like an afro-headed coward! Come to think of it, I can't even take a warrior with such stupid hair seriously!
Otsuru: Hey Jinno, I have a question. Why do you use two swords instead of one?
Jinno: With two, I can be the best, Otsuru. See, there are lots of reasons. You see with two smaller blades, I can swing faster than a guy with both hands on one big sword. And I can attack and defend at the same time. I can even fight if one of my arms got cut off 'cause I'd still have one left. It's really the ultimate fighting style.
Kuma: Why do you not draw out your sword? If you want the #1, you must first defeat me. You have come to defeat the #1, haven't you?! If you wear the #2 band, then you must fight!
Master: Jinno. Did I wake you? You should get some rest.
Jinno: Master, I smell blood.
Master: I was attacked, but that fool last his life instead.
Ninja Ninja: Afro! If you go'n run, run! If you go'n fight, then damn it fight!
Kuma: Long time, no see, Afro.
Jinno: What have you gotten yourself into? Okay, you and I are gonna talk about this later. But first, I gotta save your dumbass!
Ninja: (Praying for Brother #1) Add one more body to the body toll! May God rest this poor bastard's soul!
He's possibly Afro's equal in combat, but the giant teddy bear head worn by this mysterious mercenary does deract from his intimidating stature. That's usually not a problem once Kuma shows off his sword skills, though.
Kill Bill Vol. 1
The background music during the fight between Afro and Kuma sounds similar to "Don't Let Me Be Misunderstood" by Santa Esmeralda which is played during the fight between Uma Thurman and Lucy Liu in Kill Bill Vol. 1 which was also a samurai sword fight in the snow.
RZA, who made the music for Afro Samurai also was involved in the soundtrack of Kill Bill Vol. 1