I got hired for MNR but I assume it's mostly the same process.
The first background investigation they'll ask you to complete is employment/education. Put everything in as accurately as possible as they'll get your exact position title and work dates from your previous employer and they'll match it against what you reported. Most previous employers don't specify whether you were fired or not, reasons, etc. But still, be honest. I was honest and told them I was involuntarily terminated from my previous job (even though they didn't tell them or give any reasons), with the exact reasons, and still got hired. Being honest goes a long way with MTA, lying is a sure-fire way to end any possibility of landing a job with them.
Once you pass that stage, you'll wait a little longer until they're ready to hire more people for the position. At that stage, you'll be asked to complete ANOTHER background investigation, which checks on your criminal history. Again, be completely honest. I know a couple ex-convicts who work with me, so having a record isn't a dead-end. I had no history so it wasn't an issue, but I can't speak for others. All I can tell you is to be honest - it's a very thorough investigation and they will find out if you lied. Even if a record has been sealed or expunged, they will find out, and you need to report it.
Once that's done, all you have to wait for is your offer letter, and you'll officially be an employee for one of the biggest transit agencies in the world.
Good luck!