Wisconsin hasn't won any national championships in football, so I'll stick to non-conference champions.
The 2005 Badger team that beat Auburn in the Capital One Bowl. The Badgers had a shot at winning the conference title, but consecutive losses against two very good teams, at Penn State and against Iowa, at the end of the season ended that dream (Penn State played in the Rose Bowl; Iowa beat Florida in the Outback). That left the Badgers deep underdogs to Auburn, a team that itself just missed the SEC championship game after an OT loss to a very good LSU team (which lost the SEC championship game and crushed #9 Miami in the Peach Bowl).
This was John Stocco's junior year and Brian Calhoun's only season for the Badgers. Joe Thomas was a junior, and Owen Daniels was a senior. Calhoun was as exciting a Badger to watch as any I can remember--not the best, but fun to watch.
This was Alvarez's last team, and Paul Chryst was the OC. This was at the peak of the "the Big Ten can't hang with SEC speed" years, and Wisconsin was given "no chance" against the mighty Auburn Tigers.
There were no unforgivable losses on the season (though the 51-48 loss at Northwestern was tough to take), and the Badgers won one of the more memorable games against Minnesota, with Jonathan Casillas blocking a punt then recovering it in the end zone for a last-minute win (same day as the Bush-Push and Michigan's last-second win over Penn State:
http://www.espn.com/blog/statsinfo/post/_/id/110482/why-oct-15-2005-was-the-wildest-day-in-college-football-this-century).
Then there was the upset win over Auburn. It wasn't a blowout, but it was decisive, with John Stocco taking a knee in the red zone for the 14-point win. What I remember most about it was Chryst's inside screens to tight ends and wide receivers (Daniels and Orr) for big chunks of yards.
It was also the first Wisconsin win over an SEC team in a bowl game (and the Badgers lost to Georgia to end 2004 and Auburn to end 2003; the Badgers were 0-4 against the SEC in bowl games; they are 3-2 since).