So in the film, NYPD officer James Edwards (Will Smith) is recruited to be one of the MIB after he impressively chases an alien on foot for a couple blocks. He is recruited into a class along with other candidates from the Marines, the Navy Seals and the Air Force, all chosen because they are "the best of the best." And Will Smith gets a shot because he can run fast for short distances.
"He passes the 'have a beer' test. Let's promote him to the World-Saving Department."
But, in a "it's so crazy it just might work" moment, Edwards does get the job and becomes Agent J. Well, all right, the MIB are clearly an "outside-the-box thinking" type of organization, so maybe they spotted some innate talent in him that could be cultivated with the right training. So, he gets the job.
"I like the way you shoot children."