Depends on what level of background they do. Unless you need security clearance or integrity status (think police / sheriff) they more than likely won't contact peers, friends and family.
Microsoft and other corps do background checks at a level 1 station meaning they are just checking for convictions and validate any stated degree necessary for the position.
I would go on the other interviews and let them know you may have a position lined up. That can set you up for a harder look and maybe a pay scale bid. For example when they find out you have an offer but it potential start date is a couple weeks away they may look at your resume harder figuring is someone (hopefully a competitor) is already making you an offer, you may be a resource they don't want to get away. If they ask what your salary will be there, add 20% and see if they match it
