Which ones were actually convictions? If it's one DUI conviction and a couple of dismissed charges, that's pretty straightforward. If you've done HIMS and have a medical, should be fine for regionals. Shoot for an AA wholly-owned so you can flow to AA without further interviews. Even if you have to interview for majors, by the time you get there this should be old history. You'll be at a competitive disadvantage as far as getting an interview call, but if you can get a call you'll probably be able to overcome this at the interview.

If you have multiple convictions, that's getting a little tougher but assuming you were pretty young, you can probably convince someone that you've got your act together. You'll need to keep a very clean record going forward, and try real hard not to fail any aviation training events either.

It's getting hard for civilian employers to ask about arrests which did not result in a conviction. If they don't (can't?) ask, don't tell them.

If they ask open ended questions like "is there anything else we should know about?", talk about your passion for aviation, or charity volunteer work, or something positive. An interview is not confessional, don't answer questions they didn't ask. That's partly why they ask questions like that, hoping you'll answer questions they're not allowed to ask.