There are ways to see the Sistine Chapel (and Vatican Museums) without 5,000 other people-some of them costly, some more reasonable. The cheapest way is to book your ticket online for the first entrance of the day (8:00 am). Not only will you skip the line, but if you are a fast walker, you can get through the museum and to the chapel before anyone else. You won't be all alone in the chapel, but it will be a much more pleasant experience than waiting in line with thousands of other tourists. Another way to see the Chapel with fewer people is to book one of the after hours visits that the Vatican is hosting every Friday through June. Again, you won't be all alone, but you will see the Chapel in a different light with much fewer visitors. Both of these options cost 15 Euros (the regular price of admission) plus a 4 Euro online booking fee.
For a REAL private tour, see Viator.com for their exclusive after hours tour of the Sistine Chapel. For the relatively low price of about $365 (hey, it's the Sistine Chapel!) you can enter the Museums and Chapel privately with a small group. Other options for private tours include becoming a member of the Patrons of the Arts in the Vatican Museums. For $250 per person (under 35-$500 for over 35!), when you arrive at the Vatican you will be greeted personally and taken on a private tour of the Museums and Sistine Chapel.
At last the long entrance lines are a thing of the past!