Most employers are the ones that do the actual background check.  You shouldn't have to do anything.  But, if there is something particular they want from you, then they should be able to tell you what it is exactly and where to get it.  (For example, diploma, single certificate, etc...)

For an actual FBI background check, you have to get fingerprinted and send it in the mail along with the necessary form. It takes 6-8 weeks. I'm currently waiting for mine because it's required to get a job teaching English in Korea. However, I don't think a real one is required for a job in China. I would ask them to be more specific. They probably just mean a simple online background check, but I'm not sure why they wouldn't do it themselves.

Probably They just want check if You lie or not in Your CV (or how much)

i had a roommate at aston english in jinan last year who left china to go to vietnam and he got a background check for his time in china and ordered a background check from the state of pennsylvania website even though he lived in maryland. he used his social security number and credit card and ordered it online from china.

sorry i dont specific details but good luck to youl