Kobayashi's the man behind the man—but not the man behind the man behind the man (that would be Keyser Söze, Kobayashi's boss).
After the five suspects fly to L.A., he tricks them into working for Söze by hiring them to do a job through Redfoot. When that works out really badly—though they manage to steal the drug shipment Söze wanted—Kobayashi confronts them in person, telling them that they all ripped Söze off without realizing it, at one time or another in the past, through various jobs they've pulled.
They can either help Söze by stealing a shipment of drugs from a group of Argentinean gangsters, or they can punk out and have horrible things happen to them and to their families. When Fenster tries to skip out on the deal, he gets killed—proving that Kobayashi is for real.
Kobayashi presents these stark alternatives in a totally straightforward, businessman-like way. He's totally impersonal—it's not like he has personal reasons for wanting these guys dead, and neither does Keyser. It's just business.
We can't get much of a read on Kobayashi because we don't see him lounging around at home in his underwear or buying oranges at a farmer's market or anything like that. He's Söze's representative, his messenger—and not a multi-sided characterin any way.
After Fenster dies, the suspects think they can threaten Kobayashi into releasing them from the deal. They kill two of his guards and then put a gun to his head. Trying to remain impersonal, but clearly fearing for his life, Kobayashi tells them that he actually is working on a legal issue (an extradition) with Keaton's girlfriend, Edie Finneran, and she's in the building right now.
Keaton doesn't believe him at first, but Kobayashi shows them that it's true. Now they're scared, realizing that Kobayashi and Söze have a disturbingly long reach, and they agree to cave and attack the boat…which works out terribly for all of them, except Verbal.
And then there's the twist: in the end, we discover that Kobayashi is a fictitious character—although someone like him may have existed. After realizing that Verbal made up much of his story using details from the office bulletin board, Kujan drops his coffee mug—it breaks and we see the word "Kobayashi" printed on the bottom. Kobayashi is just the name of a coffee cup manufacturer.
But the guy who picks up Verbal at the end looks exactly the same as the Kobayashi from the story... so, maybe he actually was the henchman who contacted the five usual suspects?
But we'll have to wait for the (non-existent, never-gonna-happen) sequel to find out.