Having just seen a Kill Bill double feature, I just have to put in my two cents.
Someone already mentioned "Beyond Monks" and the martial artist class. Rather good resource, Hong's
martial artist is a similar homebrew.
But generally even the core should be sufficient if your player is willing to live with the fact that it'll take some time 'til he gets all of the Bride's power -- if he ever will... A slight modification of the monk gets some good results. Allow the Katana as a monk weapon, maybe requiring a feat or delaying unarmed damage. With its skill selection and flurry a whole lot is done already. But in the D&D universe, this char will learn that he'll attack quite a lot, but won't hit as good as some other characters.
So if you go for fighting superiority, stick with the fighter. I'd take a rogue level straight at the beginning (first level), or just use the apprentice rules from the DMG and created a mixed rogue/fighter from the very start. That should be okay for the occasional sneaky stuff, but then continue the direct route. Uma hasn't done that much hide-and-seek, she just went in and tried to kill. Tumbling is nice, though. And anyway, you'd be infringing on the areas of the rogue (and maybe even the cleric).
Go for the appropriate skills, whirlwind attack seems a definite must, specialisation wouldn't hurt either. Stick your skill points in Jump and maybe cross-classed Tumble if your Int turns out alright. Dipping a level in assassin gets you more sneak damage and the rather nice death attack. Duelist sounds nice, too, but you'll have to see whether you'll get enough feats to live with before you do that.
The major downside of applying the Bride to D&D is AC. Without some heavy metal on you, you'll just get hit a lot. Even the monk hardly ever gets too a decent enough level for hardcore fighting in the middle of a huge bunch of foes. Combat Expertise helps, even more reasong for having a bigger BAB (thus fighter and not monk). Improved Expertise, too. Without rules for parrying or a defense bonus (a la D20 modern, Star Wars or UA) you're forced to wear some protective layers (at least from my experience, maybe with higher attribute allowances a Monk has enough wisdom and dexterity to avoid being smashed down by any brute).
But well, if you're the DM you might drop enough bracers of armour and other stuff to have something less conspicious.
If if just takes too much time to reach whirlwind attack, a "poor man's bride" (hey, Hong, how's that for an euphemism?) is Great Cleave. Attainable at first level for a human fighter, this can result in a lot of dead low-level foes right at the beginning. If you want to play Sunless Citadel, both kobolds and goblins are prone to that approach. Then Dodge and Combat Expertise at the next levels, for a little bit more damage avoidance and aim for Whirlwind attack after that.
A rather unusual approach could be crafted by Unearthed Arcana's "Whirling Frenzy". +4 Str/+2 AC, one extra attack at highest bonus (and -2 on all attacks if that is used) instead of the usual Rage benefits. Actually, the Barbarian isn't that far off. She could certainly take a lot of damage, so the d12 helps, and heck, if ever a woman was angry... Plus 4+Int skill points, enough to max out Jump, Intimidate and Tumble as cross-class.
