My advice comes from 11 years teaching in mainland China. All the terms below I see on all my documents, so nothing is made up.
The legal route is to apply to a prospective employer while you are outside of mainland China and when you are accepted, you then take the invitation letter to the consulate in your country to get the "visa" ("Residence Permit for Foreigner in the People's Republic of China").
This is not a good place to ask about ESL requirements. Half the answers you get will be from people who either have no RECENT first-hand experience or did something dodgy and just managed to get away with it, so they come off suggesting you do the same. Try Daveseslcafe. A higher percentage of the answers there (but no guarantee) will be correct. The State Administration of Foreign Experts Affairs issues the certificate for teaching in mainland. The police issue the Registration Form of Temporary Residence. You cannot get the latter until you have a proper visa. You cannot get remain in mainland, until you get the latter or have been set up with on-campus housing. If you have no Registration Form of Temporary Residence on file with the police and no residence on-campus with your job, and you are questioned, you will be fined up to 600 RMB if you are otherwise in good standing with a proper visa and Foreign Experts Certificate. If not in good standing, you will be deported.
The visa and certification rules are much more strict now than before 2005. Hong Kong visa is not nor ever has been a legal work visa for mainland. People assume it is, because they get past the entry officials with it. Entry officials do not care about your mainland situation. They care about who is permitted to enter, and that includes those with Hong Kong work visa WHO ENTRY OFFICIALS ASSUME ARE JUST COMING IN TO VISIT. If you want to stay in mainland more than 30 days on a Hong Kong work visa, you could be confronted by the police and the likely result would be exportation and no chance of ever getting a proper visa for mainland in the future.
The current rule is that if you are in mainland no matter how you got in, and want to get a proper work visa (no longer called visa as such, but rather called "Residence Permit for Foreigner in the People's Republic of China"), you must return to your home country and apply at your local consulate. This is how it is now even for those who came in by way of Hong Kong work visa. Your employer does not issue the visa. Neither he nor you can side-step the proper application process.
All of the above applies to ANY AND ALL types of jobs you may hope to land, no matter what level of school, no matter if school or company that offers English class to employees. If any employer allows you to work without a "Residence Permit for Foreigner in the People's Republic of China", he is in violation of the law. That's why he may pay you in cash only (no bank auto-deposit) and keep no records on the work premisses (even if you signed documents). He will not admit ever having seen you before if questioned by the police.