A closed completed case would not appear on a general background check. But if its a check by a government agency or for a regulated profession like medical, day care, teaching, etc, it will appear because that type of check is much more thorough. Also, prospective employers regularly hire companies that look at all criminal records at the courthouse and therefore can easily find open first offender cases like yours as well.
Unfortunately, what often shows up on employer background checks is the arrest record from the GCIC statewide law enforcement database. Unlike convictions, which can be expunged from a person's record, an arrest will always show up on a thorough background check using the GCIC database or other local, state or federal law enforcement databases.
Typically, in a pre-trial diversion or first offender case, the charges are often "dismissed" after the individual has met certain court ordered criteria such as completion of community service or the payment of a fine. However, whether such a disposition shows up on a law enforcement records check is a more complicated and uncertain question.
This answer is meant to be general, educational and informative and should not be construed as legal advice. Depending on the specific circumstances of your case, I would consult a knowledgeable attorney in your area.