Kobayashi Maru is the name of a specific no-win test in the Star Trek world. The test is meant to test the cadets on what they would do in such a no-win situation. The test is named after the 'Kobayashi Maru', a stranded space vessel that plays a central role in the test.
In this episode of The Office Dwight interviews several people for the position of new part-time salesman. In the interview with Clark, who Dwight does not want to get the job, Dwight sets up a similar no-win situation. Clark recognizes this, and his response is I'd Kobayashi Maru it (the situation). So yeah, Clark uses a noun (the name of the test) as a verb. We can do this in English and that's one way new verbs are formed.
As for the meaning, it's a bit obscure at first. Clark could plausibly mean something like I'd apply the methods of solving the Kobayashi Maru test to it. But this is just trying to put some meaning to his expression. What's important at this point is to realise that whatever Clark meant, Dwight (a super geek Star Trek fan) is impressed by its usage, since it alludes to the Star Trek test.
Clark uses the same phrase (to Kobayashi Maru something) a second time, and this time the meaning is clear. The interview continues with the following lines immediately following those in your question:
Dwight: Damn it! Perfect answer, again.
Dwight: Think Dwight, think. You have a ream of 16-bond …
Clark: You know what, Dwight?
Dwight: And anoth—
Clark: This interview’s over, and I get the job. I just Kobayashi Maru’d the whole process.
Clark: Yeah. Star Trek rules.
Dwight: It does, but still no.
Clark: Come on, man. I mean, did Trevor do that? Did Rolf do that?
Here, Clark uses the phrase in a similar way as the phrase to deep six something. You can clink on that link for Oxford Dictionary's definition, but the Wikitionary definition that user3169 links to is easier to apply here:
(idiomatic) To discard, cancel, halt; to completely put an end to something.
In Clark's second usage
This interview’s over, and I get the job. I just Kobayashi Maru’d the whole process.
Clark clearly uses Kobayashi Maru as a synonym for deep six. You can insert deep six'd for Kobayashi Maru’d with no change in meaning.