Well, if you want a "real" traditional samurai, your PC would focus on archery first, then mounted combat, then bladed weapons, in that order. Samurai were archers first, before anything else. In fact, they only used the blade to behead enemies, and close in fighting was a last resort. They really didn't start using it as a primary way of fighting until they fought the Mongols. And that wasn't just katana, but naginta and spears equally.
It's one of those American made myths we have, like the ninja. Ninja in history didn't wear all black jammies. They dressed in whatever clothing made them blend in. And shuriken were never designed to kill. In fact, there's no historical record of one every killing anyone. They were used to distract while you closed in and killed with your blade. There are tons of examples of how our Americanized takes on history are totally wrong, but oh well. They're cool.