From 1946 through 2004 there have been 59 Division I-A Football “National Championships.” According to the NCAA Record Book (page 89), 77 teams from 30 schools have won or shared in the 59 championships (14 times there have been co-champions, twice there have been three teams named).
Fourteen schools have won or shared the title more than once:
Oklahoma (6 outright, 1 tie)
Notre Dame (5 outright, 3 ties)
USC (4 outright, 3 ties)
Miami (4 outright, 1 tie)
Nebraska (3 outright, 2 ties)
Alabama (2 outright, 5 ties)
Ohio State (2 outright, 4 ties)
Texas (2 outright, 1 tie)
Penn State, Florida State, Tennessee (2 outright each)
Michigan State (1 outright, 2 ties)
Michigan (1 outright, 1 tie)
LSU (2 ties)
These 14 schools account for 61 out of 77 championship teams (79%).
I would argue that to win a national championship you have to schedule (and beat) at least one of the 14.
Seven schools have won the championship once. They are Maryland (1953), Syracuse (1959), Pittsburgh (1976), Georgia (1980), Clemson (1981), BYU (1984) and Florida (1996).
Nine schools have been co-champion once, but have never won the title outright. They are UCLA (1954), Auburn (1957), Iowa (1958), Minnesota (1960), Mississippi (1960), Arkansas (1964), Colorado (1990), Georgia Tech (1990) and Washington (1991).