You have a couple of options if you want to become a football scout: be a former coach changing to the scouting side of things, or get into an internship program. If you have a connection to football, you're in a great position; the rest of us have some work to do.
Get an unpaid internship with an Arena Football League team, arena 2 franchise, Intense Football League, United Indoor Football or any number of minor leagues. Even if the team doesn't have an internship posted, call or stop by the front office; these leagues have tight budgets and usually can use help.
Read up on coaching techniques and drills, and talk with coaches.
Get a coaching job with a high school football program's freshman team. Some schools need reliable people who are willing to make the commitment for the season, so simple dedication is usually the only job requirement.
Send out your resume to every NFL, AFL and CFL team. You probably will get a lot of rejections, but your name will be out there.
Attend events where you can meet the decision makers; the most notable one is the NFL Combine. You won't need a pass to get into the official activities, and you can hang out in the bars and restaurants for a few days.
Take a full-time internship with one of the minor-league franchises. If necessary, find a job with flexible hours that will allow you to continue to coach in the fall and hold an internship in the spring. As you gain experience, take advantage of any opportunity to move up--get promoted from freshman to junior varsity to varsity or from intern to unpaid scout.
Keep building your network. Find any way to get your foot in the door of an NFL team. Teams hire literally dozens of unpaid people to help out on game days, doing anything from passing out stats in the press box to pulling camera wire on the sidelines. While these jobs have nothing to do with scouting, each of these positions comes with an all-access pass to the stadium and an introduction to an NFL staffer. You never know who could become your advocate.
Keep going. Trips to the combine, working as a coach and unpaid scout will get you there eventually (be prepared for it to take several years).
Once you have a bit of experience, list your resume on Teamwork Online. You can also check for jobs on Football Scoop, Jobs in Sports and Work in Sports (see Resources). If you get an opportunity to get a position with an NFL team in any department--ticket sales, marketing or public relations--do so. While it's not what you're looking for in the long-term, it gets you into the building.
Be prepared for this to take years.