According to the official novelisation, when you join the MiB you simply remove yourself from your old life. One assumes that a general cover story is given to work colleagues ("I heard he went to work for the FBI or something") and that any close friends/family members are neuralysed to stop them from coming to find you.
He looked at Edwards. "Not that you're gonna have to worry about
people who don't know our little secret. Thing is, if you join us, you
won't have an outside life. No wife, no kids, nothing. You cut off
contact with everybody you know outside the Black. In your case, it
shouldn't take too long. And what you get for a prize is long hours,
dangerous days, and no recognition. You don't even get a favorite
shirt—unless you like white."
MiB: Official Novelisation
That all being said, it would appear that not having close ties/family to miss you, etc. is a positive selling point for any potential MiB candidate.
And to lower the chances of anyone bumping into you
Behind him, Zed said to the kid, "You'll conform to the identity we
give you. You eat where we tell you, live where we tell you, and you
get approval for any expenditure over a hundred dollars. In writing.
In triplicate."
MiB: Official Novelisation
and
Kay said, "You just quit the force, paid your rent and gave up your
lease—"
"Man, that's a rent-controlled apartment!"
"—you don't exist at the DMV any more, you never got a library card, a
passport, or a season ticket to see the Yankees. Your credit card
records went away, and your schools have no official records of you
ever attending them."
MiB: Official Novelisation