There is no surefire formula to upgrade your seat at an airline. However, there are several factors that could work in your favor. Upgrading depends on your status, the status of the authority you interact with and the capacity of seats.
If possible, see if upgrades are available further in advance. Show up to the airport early. Talk to a reservation agent at least 24 hours in advance to see what the odds are in getting an upgrade. Ask how much an upgrade will cost at the last minute, sometimes deals are available if you ask. Make friends with the ticket agent at the counter. Ask them about themselves and then let them know a little about who you are. If you're a V.I.P. or C.I.P., Commercially Important Person, subtly let this be known. You may receive an upgrade if you let them know you're there for a special occasion, such as a honeymoon. This may be tricky, since you do not want to come off as boastful. When stepping onto a plane, if you see vacant 1st class seats, kindly ask your flight attendant if those seats are taken. This could be your chance at scoring a truly last minute upgrade.
Airlines give preferential treatment to those who purchase full-fare. Also return customers are favorable, especially if you've maximized miles on just one airline. Some customers give up free flights from frequent flier miles in an exchange for an upgrade.
Seek elite-level frequent flier status and you're heightening your chances for an upgrade. Consider the following acronyms of elite status and see where you fit: V.I.P., C.I.P., P.N.R. -- Passenger Name Record which means the reservation carries a "pre-authorization for upgrade", S.F.U. -- Suitable For Upgrade, or "do all possible to assist". This should place you in advance to a position for upgrade. Just like in any situation, looking the part helps a huge deal. Rather than choosing sweats and a t-shirt, consider dressing well, much like a stereotypical V.I.P.
