Source of information for this is a CBS article written by Pete Tiernan.
In order to winnow down your list of possible champions, you can use the criteria Pete Tiernan set forth in his article. I'll run down the list here for you with this season's stats:
Criteria One: A top 4 seed, score over 76 ppg and beat opponents by 10 points on average.
The "top 4 seed" requirement eliminates 75% of the field automatically (only a 6 or an 8 seed have ever been the exception). If you are playing the percentages, this helps out tremendously. The scoring & scoring margin requirements knock out 1-seed Louisville, 2-seeds Michigan State and Memphis, 3-seeds Villanova and Syracuse, and 4-seeds Xavier and Washington.
Historically, 22 of the 24 champions have had these attributes. This narrows our list of possible champions to Wake Forest, Kansas, UConn, Missouri, Pittsburgh, Duke, North Carolina, Gonzaga and Oklahoma.
Criteria Two: Been to the previous tournament
This eliminates Wake Forest and Missouri. 21 of the 24 champions have had this attribute in addition to meeting the first criteria.
That means by using those 4 measuring sticks, you could whittle this year's possible champions down to a list of 7 teams: Kansas, Connecticut, Oklahoma, Gonzaga, North Carolina, Duke and Pittsburgh. Notice that the Sooners are on the list, despite being picked everywhere as a big upset pick. Additionally, based on the history of the tournament, you could be about 85% confident that the champion would be from those 7 teams.
Of course, this sets up many problems in your bracket when choosing. 1/7 is still only about 14% odds. If you want to whittle it down further, you could get it to 5 teams with:
Criteria Three: Increase scoring to 77 ppg, increase scoring margin to 11 ppg, be from a "big six" conference, and have a coach that's been to the tournament for at least 5 trips.
70% of the champions and 16 of the last 18 have met those criteria. This eliminates Oklahoma and Gonzaga. So it is still fairly likely that the champion will be from Kansas, Connecticut, Pittsburgh, Duke or North Carolina. You would have about a 20% chance of correctly picking the champion.
In any event, those little nuggets should help you decide which teams to advance far in your bracket. Be warned though that usually 2 or 3 of the original list of teams will go down earlier than you expect, sometimes even from the pool of teams after it's been whittled down for awhile. It's not an absolute way to determine the champion, but it's a good way to make an educated guess.
Also, a warning, the pool of teams you find is also not guaranteed to make the Final Four, although usually 2 or 3 do. These criteria help you find the strongest teams in America, so it's more likely that those teams will find their way to the Final Four.
Do what you want on your bracket, but consider picking Pitt, Duke, UNC, UConn or Kansas as a champion. And don't eliminate those teams, or Oklahoma, Gonzaga, Mizzou or Wake Forest too early.