Unless you have received a full pardon, if you have a criminal record, go back home and stay home. Teaching is not for you. If your record was expunged it is not normally available to anyone and it will yield a sealed file hit if anything at all, on a computer look up depending upon the jurisdiction, the type of crime(s), the date of offense(s), date of conviction(s), end date of sentence, and other related issues. In fact these days it is hard for anyone convicted of any crime to even clear a passport application. The new passport applications at least for the USA, ask if you have warrants pending. In fact, the US Marshal's Service will check those in addition to the IRS, HHS, and other checks. As of now there is pending legislation that will block you from getting a passport if you owe back taxes which will most likely pass. The US Consular section of the US Embassy, to include the average clerk, (at least some of them), will soon be designated as a law enforcement agency for the purposes of accessing data bases that police need access to as part of an ever expanding data file sharing system. If you go to your embassy for any reason and if you have warrants they will seize your passport and cancel it, or amend it so that it is only valid for return travel home. Depending on what state you come from, a record being expunged is not necessarily the same as a full and unconditional pardon.

China does in fact check criminal history when they run your passport and scan it for a new visa or when you enter and exit the country, however as it is a non-extradition country with the USA, they won't hold you. But they can refuse you entry.
However if you go to HK, they will arrest you and hold you for extradition from the USA, assuming that the USA wants you bad enough. Normally, they'll just kick up a few clowns from Homeland Security to fly you back to the states from HK since they are already there in HK anyway.

With INTEPOL data sharing among many nations, don't think that they cannot find you if they really wanted to. As for your record(s), if you are in China, China knows about it and in fact, they knew about it at the Chinese Consulate when you applied for your visa because they do check those things and they have access to a number of shared data files on a need to know basis.

In my case I have a perm residence card for China as do a few of my friends and they do fingerprint you for that and the prints are cleared through the FBI and INTERPOL. The local branch of the foreign affairs police will handle that since they speak English and they network with foreign law enforcement daily.

I have seen a few foreigners being denied entry into China at the HK border and I have also seen a few foreigners being arrested at HK going into HK from the mainland out of GZ.

Depending on where you work and who you work for in China, yes a criminal conviction or an entry on file, will in fact stop you from working in education and rightfully so.

I hope that answers your question and gives you some more useful insight.