For the locality you're currently in, it's relatively easy - just go to the police station and inquire. (And as a matter of fact, the police station here said go see the "safety officer" at your school, who turned out to be an entire department of police officers stationed on campus...)
But what if you've been places, man, seen things, been employed in other provinces. The "safety officer" says they can provide a background check extending only as far back as my current stint of uninterrupted employment. In particular, they can't provide background for my two earlier periods of employment even though those periods were at this same school in this same location.
So, is there any recognised procedure for obtaining a complete Chinese criminal background check that will cover a foreigner's entire time in China from beginning to end , even if they've changed employers and even province a few times?