The background check only happens after you admitted and decided to go the school. So that happens very late in the process ( probably a couple of month before class starts) . Not all schools are doing background check.
Suddenlyisee, I don't know for your company, whether your HR would tell your boss, but eventually you have to quit for your MBA. Before the background check process, you would receive full info about the process and verify certain info for Kroll before it starts. So at that time, you can consider quit your job to aviod awkwardness with your boss.