Criminal background checks are a necessary part of being hired. They are not meant to offend or to invade privacy, merely to assure the safety of employees, customers and clients.
Follow accepted practices for writing a business letter. Make sure everything is left-justified, for example, and use formal address.
Make your language clear and concise. This is not the time to use a lot of adjectives and adverbs or to hint at a subject.
Check your background yourself to make sure there are no mistakes. Remedy any mistakes as soon as possible. Also, you can ask to have any one-time-only offenses expunged after five years, unless you are a sex offender.
Be honest. Explain the circumstances behind any offenses on your record. A full disclosure will allow your employer to see exactly what happened and how you have changed since then.
Research the positions you are applying for. Find out what information your potential employers will need about you so that you can be prepared to explain.
Maintain a positive, open attitude, especially on paper. Often things that may be funny when said are not so funny when written down.
Remember that this is not meant to be an invasion of your privacy.