Join a frequent flier program. Frequent flier programs offer complimentary upgrades for members who accrue a certain number of miles. Fliers in select tiers are often given priority on upgrades, and airlines may "bump" other passengers from first class seats who are not in a higher tier. Airlines also offer free coupons to fliers who spend a minimum amount on their airline credit cards.
Offer to get "bumped." Airlines often overbook flights (especially during peak travel times) in order to ensure all of their planes are full. During these busy travel times, if all passengers show up for their flights, reservation agents will ask passengers to volunteer to take a later flight. If your travel plans are flexible, volunteer to take a later flight if the airline offers to reward you with an upgrade (and sometimes a free airline ticket).
Purchase a Y-up fare. Y-up fares are a mysterious "insider secret" that most fliers don't know they should ask for. These fares are booked through airlines as coach fares yet are booked as first class seats. Often, they are less expensive than coach seats, and a traveler holding one of these tickets will not be "bumped" back into coach by a traveler in a top-tier frequent flier program, either. These flights are most frequently booked with last-minute and/or emergency travel. To purchase a y-fare ticket, ask your travel agent to book it or use an online y-fare search engine such as farecompare.com.
Pay the first class fare or buy an upgrade. It's no fun paying full price for a first class ticket, but, if you have the means to do so, it's the easiest way to fly first class. Many airlines offer upgrade vouchers for a few hundred dollars that allow fliers to upgrade a current coach ticket. Neither one of these options is cheap, but they are less of a hassle than upgrading through a frequent flier program.
Have a good attitude. Follow the old saying that you can attract more flies with honey than vinegar. Simply having a good attitude is going to help you get that first class ticket. Most ticket agents deal with ornery or unhappy customers and are relieved by a customer who treats them with respect. While this is not a sure-fire way to get that first class seat, you can almost guarantee that a bad attitude will be your one-way ticket back into coach.