How hard is the LAPD background check?

There are no tips for passing a background investigation as they are investigating things that have already happened in your past (hence, background). However, do you know if they administer a polygraph examination? That will be your biggest hurdle. If they do, BE HONEST. Be as honest as possible, even stupid things like you took a pen from your old job. They will immediately disqualify you for any deception. Traffic accidents happen, they are accidents. They will most likely not count against you unless you have an extensive record.