The Packers have been underdogs 40 times since Rodgers was the starting QB, and they're 15-25 in those games. That includes games in which he did not play though, which I'll try to take out of consideration.
Here's the link.
2008 8 (2-6)
2009 4 (2-2)
2010 5 (3-2) (Rodgers didn't play vs. NWE - L)
2011 1 (1-0) (Rodgers didn't play vs. DET - W)
2012 3 (1-2)
2013 6 (1-5) (Rodgers didn't play vs. PHI - L, NYG - L, DET - L, DAL - L)
2014 4 (0-4)
2015 4 (2-2)
2016 5 (3-2)
So that makes the Packers 14-20 in 34 games in which Rodgers started as the underdog, which ranks 7th best winning % in the league as the underdog.
In that time only New England (21) has been an underdog fewer times than Green Bay.