You may be able to request your records from each state in which you have ever been charged, and then send a copy to the place that is denying you housing to prove that you were not convicted. At the same time, call a lawyer and see if you have a case for a civil lawsuit for libel and sue the background check company. You ought to have grounds for damages including at the very least the cost of temporary housing while this gets cleared up, plus legal and administrative fees incurred in getting copies of your records. Get the lawyer your proof and estimates of your losses incurred due to this (lost wages from time off work to deal with this may also count), and let him do the rest. Call multiple lawyers to find out their fee structure to make sure you receive full amount of the monetary damages you suffered after he takes his fees, and also go with one who gives a reasonable timeline on dealing with it.
In the meantime though, getting your records straight including getting copies of your police records ought to be a high priority because this probably won't be the last time this happens and it will help if you have all your records already in your posession next time so you can refute the bogus background check results immediately.
Oh yea, last go to dave ramsey's web site and follow his plan so you can get strong enough financially that you are never at the mercy of an apartment background check. Anyone can do it on any income level, and the end result is that money simply ceases to be an issue. His plan followers are typically completely financially independent and debt free after 7-10 years, without a single financial gimmick or relying on strange investment strategies.