If you look on the website of the company that you indicated is doing the background check (http://hireright.com/Background-Checks.aspx?apsi=0) you will see that they include criminal record searches as part of their process. Thus, they have a pretty good chance of detecting this part of your background.
So, it makes sense to me to get out in front of this, rather than just hoping they don't find it. Tell the hiring company about this now.
In the many years I've been interviewing potential hires, I'm always far more favorable when people are forthright and tell me about potential red flags in their background, rather than waiting for me to discover them on my own and confront them. I don't like to see interviewees try to "sneak one past me". Past problems I can deal with. Present sneakiness (or lying) is far more problematic.
You won't always get hired when you tell potential employers about these issues, but in my experience, you'll have a much, much better chance when the news comes directly from you (with your explanation about how you have changed things), than if the news comes from a third party.