As a huge football fan, here are the NFL games that will be in town during your stay and my comments (rough estimates based on the time frame given):
Chicago
Note: The Bears aren't expected to be exceptionally good this season. They are in a bit of a rebuilding phase right now. Most people expect them to be a bottom 10 (out of 32) team. That said, this is the NFL. It's extremely unpredictable. The Bears do have some talented players that could make a strong showing and turn the team around faster than expected.
The Packers-Bears rivalry is one of the most storied rivalries in the NFL and this is usually a pretty good game. The Packers are currently one of the league's elite teams and are led by the league's best quarterback, Aaron Rodgers. Their offense is extremely exciting to watch but their defense is about average and is fairly bland overall. This could be a pretty good game but it could also easily be an ugly blowout in favor of Green Bay.
The Cardinals are a hard hitting defensive team. If they keep their quarterback, Carson Palmer, healthy, they could be a strong contender for the Super Bowl but that's a giant 'if'. This early in the season, Palmer should be healthy and ready to go. This could also be a good game.
Over the last 10 years or so, the Raiders have been one of the worst teams in the NFL. They lost the Super Bowl at the end of the 2002 season and haven't had a winning season since. Some people are fairly high on the Raiders this year but I don't see it. If you want to see a Bears win, this is probably your best bet. That said, this isn't a guaranteed win, either, so you'll still likely get a decent game.
Dallas
Note: The Cowboys are one of the most historically successful teams in the NFL. They have won 5 Super Bowls, second most in the league, behind the Pittsburgh Steelers (Ha ha! Eat shit Cowboys!). That said, they struggled for most of the past decade but they seem to have turned it around last year and played exceptionally well. The Cowboys should have one of the strongest offenses in the league this year. Their offensive line was historically good last year and will probably be even better this year. Last year, they ran an old fashioned ground and pound offense built around running back DeMarco Murray. Unfortuanately, Murray left in free agency last season and the Cowboys are left without a strong running back this year. They are likely to air it out more this year. That shouldn't be too much of an issue, since quarterback Tony Romo and wide receiver Dez Bryant are two of the best in the league at their respective positions and are extremely exciting to watch. Their defense is average at best but they compensate for it by controlling the clock on offense, so you shouldn't see too much of their defense if things work out for them. The Cowboys play in AT&T Stadium, the largest stadium in the NFL. The Texas football experience is always a sight to behold.
Over the last 15 years, the Patriots are the best team in the NFL and are last season's Super Bowl champions. Unfortunately, there's a decent chance that their quarterback, Tom Brady, will be suspended during this game. Typically, New England is a well balanced team, being above average on both offense and defense. If Brady plays, this could be the best game you have an opportunity to see. That said, this game will likely be fairly expensive, since Dallas is one of the most expensive teams to see and New England coming into town always drives the price of games up.
This should also be an exceptionally good game. The Seahawks aren't a historically successful team but they have become one of the league's elite over the past three years, winning the Super Bowl at the end of the 2013 season and losing the Super Bowl at the end of the 2014 season. They run a very hard hitting defense in combination with a ground and pound offense. Offensively, they are led by quarterback Russell Wilson, one of the best young quarterbacks in the league, and Marshawn Lynch, an exciting running back that loves to bowl through defensive players. Defensively, they are led by cornerback Richard Sherman, safety Earl Thomas and safety Kam Chancellor. Sherman and Thomas are both exceptional ballhawks that are skilled at intercepting opponents passes and Chancellor is an exceptionally hard hitter that is always good for a few exciting plays a game.