Most private schools will use a service to do background checks on any potential new hires. They will not run the background check until they have interviewed you because background checks are expensive. That's another reason why you need to be truthful about anything in your resume which will arouse probing questions and nix your chances of even being interviewed.
What the background check really examines is your criminal and credit history. If you were charged with a criminal offence or have bad credit, that will show on your background check. Take the offensive and point out that your credit was destroyed by huge medical bills as a result of your mother suffering from Alzheimer's. Don't be creative. Just tell the truth.
Part of the background check is verifying your employment history. Don't leave anything out. An employer will look askance at frequent job changes. Staying at one school for 5 years is good. Working at 5 schools in 5 years is probably not a good thing unless, of course, you were being moved around by your spouse's employer or were doing substitute teaching while you raised your children. Explain that on the application as a note or an addendum if there is no space for your explanation. The person who is screening the application will consider that explanation as she determines whether to reject your application or proceed to an interview.
Schools will want to see certified copies of your transcripts and/or your actual diplomas. Be accurate and truthful about your credentials. Schools pride themselves on having highly-qualified teachers. They cannot afford to make mistakes hiring somebody with false credentials. As a result they will call the institutions you claim to have degrees from and verify those degrees and your coursework.
Being Fired/Terminated/Laid Off
You need to explain gaps or departures before the end of a contract. For example, if you were teaching at a school and suddenly took sick and had to leave your position, that gap in your employment will be easy to spot on your employment application. Be proactive and address it before your prospective employer does. A brief explanation on the application will suffice. Something like "Resigned due to illness" or "Resigned because spouse was posted overseas" will get your application past the initial screening in most cases.
If you were fired, be positive in your explanation. Try not to dwell on the experience in a negative light. Nobody wants to hire a loser. On the other hand if the interviewer can see that you have sterling credentials and that you simply had a bad experience with a psychopathic employer, you will stand a chance.
References are an important, indeed, deal-breaking, part of your employment application. A strong reference will address things like gaps in your employment history or frequent job changes. The reference will be called by your future employer if you make the short list.
He or she needs to proactively address issues. He needs to explain that "Sally has spent the last 5 years moving from one city to another as her husband kept getting new job assignments with his major multi-national employer. She always managed to find employment but it looks like she couldn't stay in any one position more than a year. Her husband recently won a head office position and they expect to be in the Boston area for the next few years." Hopefully the reference is somebody who would take Sally back in an instant if she were available.
Discuss the strengths and weakness of your job application candidly with your references. Don't assume that they will be proactive. Make sure that they are.
Beware of the reference who merely says something bland like "Sally will be an asset to any organization." That sort of damning with faint praise is worthless. Far better is something like "Sally is a gifted Spanish and Italian teacher who shares her love of Romance Languages with all her students and colleagues. She also was the captain of her debating team at Vassar. She coached our debating team at Holy Name so successfully that they won the state championship."
Finally, be aware that you cannot be dismissed for being less than truthful on a resume. But it will come back to haunt you if, for example, you claim to have been a speaker at the Romance Languages Conference when in fact you merely served on a discussion panel.
On the other hand you can be terminated if you falsify a job application. Don't even go there. Deal with any less than pleasnt facts in your application in a positive, truthful manner. Hide nothing.