Decide if you want to watch NFL games for free or if you mind paying a fee. There are advantages and drawbacks with both. Watching football games for free online is obviously great because you're not paying for it, but usually you're relying on other people to stream the games from their TV via their computer and the quality might not be great. You're also at their mercy if they for some reason terminate their stream in the fourth quarter when you're team is making an amazing comeback to win the game.
Pay for a week or season pass to watch live NFL games if you want to avoid all the hassle. Yahoo! has teamed up with the NFL and gives you access to all games (see Resources) with NFL Game Pass, regardless of what networks they air on. You can watch any game live, except if you're in a country where they already show some NFL games on TV. The games that are televised abroad will be available online within two hours after they are finished.
Watch the reruns whenever you want. If you live in a country where watching the games live is hard because of the time difference, Game Pass lets you watch any game from the current week up until Wednesdays at midnight. When you watch the archived games you can even skip through the commercial breaks and get through games quicker.
If you only want to follow your favorite team and don't care about the other games, you can buy a team pass that let's you watch all 16 regular season games that your favorite team plays. A team pass is cheaper than buying the season pass.
If you'd rather save your money for something else and just want to watch a game or two every Sunday you can watch games for free that are streamed by people in the U.S. (see Resources). This usually requires that you have to download a new media player that you've probably never heard of before, but they are usually free as well. SopCast and TVUPlayer are two examples and they can be downloaded from, or at least you can find links to where to download them from, on the site where the list of streaming games are.
Buy a Slingbox for someone you know that lives in the U.S., or sweet-talk someone who already has one. With a Slingbox you can watch all the channels the person who has the Slingbox has access to, but from anywhere in the world. The bonus of taking this route is that you can not only watch football games live for free, but any other show or channel your host has access to.
The last option is to download the entire game after it has been played, but you obviously are not going to see it live and there's never any guarantee the specific game you want to watch will be available for download. You can usually find at least two games to download each week on Piratebay.org (see Resources).