Find the apartment complex's identifying information, including the name, if it has one, and address. The police can enter this information into a database to search for records, so the more information you have, the easier it will be for them. Be aware that a complex with multiple buildings might have multiple addresses.
Determine how far back you'd like to look. You may legally ask for every record ever taken about an apartment complex, but that will complicate and delay your search. Creating a time window, such as "in the last two years," will simplify the process.
Determine what agencies you should contact. In most cases you'll simply need to contact your local police department. However, there may be circumstances when you'll want to check with the county sheriff to make sure they've never had contact with the complex. Rarely should you need to reach out to agencies with wider jurisdictions than the sheriff.
Request the police reports by writing a polite, professional letter to the appropriate law enforcement agency requesting copies of police reports involving a specific apartment complex with a specific address or set of addresses within a specific window of time. Make sure to include your name and contact information.
Follow up on your request. Officials will usually be able to say when they expect to fulfill your request. If several weeks or months pass without word from the law enforcement agency, it's appropriate to follow up with a phone call or another letter reminding them of your original correspondence. If officials continue to ignore your request, see if your state has a public access counselor responsible for making sure public record laws are followed.