Being a member of an artist's fan club often gives you access to special tickets, ticket pre-sales, and contests. Joining may cost money, but if you really like the group, this will give you a good chance of getting passes.
Concert venues often have VIP groups as well. If you plan on going to a lot of shows at the venue, the VIP packages can be well worth the price. Along with better parking spots and good seats, they often offer access to special areas or contests for backstage passes.
Radio stations that play an artist's music usually have free tickets and passes to distribute before the show, so be sure to listen in the weeks before the concert. Call in for every giveaway, and keep re-dialing if you get a busy signal.
You can also try calling the DJs, even when they're not running a contest, and asking. Sometimes a great story from a huge fan will win tickets or passes "just because".
Whether they're through the radio station, the artist, the venue or somewhere else, enter every contest you can.
If you work for a company that has connections to the artist or the venue, see if they can get passes. If you know someone in the concert business, ask them if they can help.
If you're a writer or photographer, ask your employer if they'd like pictures or a story about the event. Freelancers, don't be afraid to shop your skills around and see if anyone needs some extra coverage. Press passes are easy for media outlets to get, and they'll get you great access.
Fans without backstage passes will often hang around at the back entrance to the venue--where the artist will enter and exit--or near the tour buses. It could be a while before anyone shows up, but many artists will take the time to sign some autographs for those waiting.
If you do wind up with passes, act cool, calm and collected. Meeting your favorite artist can be an awesome experience, but there's no need to act like a crazed, screaming superfan. If you can stay calm, having an actual conversation will make a great memory.
Don't try to sneak backstage. If you get caught, you could get kicked out of the show.