To make a volunteer opportunity require a background check of a certain level in order to sign up, go to the volunteer opportunity record and scroll to the restrictions section.

There you'll see a new field labeled "Min Required Level of Verification.

Set the Min Required level of Verification to Level 1, Level 2, or Level 3.

Save the Volunteer Opportunity Record.

Now, only volunteers who have the required level of background check will be able to sign up for the opportunity.  If they do not have the required level of background check, they will receive a message directing them on how to apply for the background check.  

See this article for how the volunteer experiences Verified Volunteeers on the public site.