There's a summary of the background check process in the AOS Process Spreadsheet (see link to spreadsheet in the first post on Page 1 of this thread - once in the spreadsheet, look for the tab titled Background Check).
Background check timing will be different for everyone, I assume anywhere from weeks to months. Some checks can only be completed once they've taken your biometrics too. If the background checks reveal anything they want to investigate further, the process will obviously take longer. I believe they can check overseas records too.