Bankruptcy would erase any money that is still owing because you broke the lease. But frankly, BK over what is likely a pretty small amount isn't a great idea, and many creditors are accepting south of 20% in full and final settlement of outstanding debts.
BK also will not erase the eviction public record from your credit report (however, it would allow you to vacate the judgment, which would then appear "vacated" on your credit report). Your real problems are Saferent, Choicepoint and the Unlawful detainer registry.
It means you will be stuck renting at individual landlords unless and until you can get out of those systems.
If you have the financial ability to do so I would recommend that you contact the creditor and try to resolve this with them. As stated previously, many creditors are now accepting quite a bit less than 100% to settle. Unless you have other outstanding debts, it would make more sense to resolve this than to file bankruptcy.