Keep your eye on the prize: elite status. Airlines will give priority for upgrades to top-tier members. Pick an airline and stick to it. If you are a regular customer you are more likely to get those three coveted letters: SFU (Suitable For Upgrade) next to your name on the passenger list. Also, there are often rewards for accumulating miles quickly (usually over one calendar year) and different "perks" are awarded each time you hit a certain mileage tier.
Points systems are not born equal. For example, with Air Canada, you can only use points for complete bookings in economy or business and cannot use points to upgrade from an economy ticket. If booking on Qantas, you can buy an upgradeable economy ticket and request for an upgrade to premium economy or business. British Airways, on the other hand, makes it nearly impossible for those who fly economy short-haul flights to ever earn enough points to make it to the next level.
Obviously, this might not be possible! But it's worth bearing in mind that if you are travelling as part of a small group, an airline may not be able to upgrade everyone and, therefore, won't upgrade anyone at all.
Checking in late means the economy seats are likely to have been filled, meaning you may get booked directly to business class. This is a very risky strategy, of course, as you chance not being able to get a good seat in economy, or getting split up from your travel partner.
Pick a flight that will be using a plane with a large first class cabin. You can find out this information from sites like Seat Guru.
Have time to spare? Every so often an airline will oversell the flight and will need volunteers to give up their seat. If you don't mind the delay, you can score a flight voucher and/or a free upgrade certificate (they've done this before on Cathay Pacific), and, before long, could find yourself happily snoozing in first class.
If you purchased a full fare ticket and travel on an oversold flight, then you also have more potential for a courtesy upgrade.
This won't be the reason why you get an upgrade, but not being dressed suitably could be why you don't. Take Victoria Beckham - we reckon she's never travelled economy in her life.
Once upon a time, being nice could result in someone at the check-in counter upgrading your seat. Nowadays, stories like that seem more like an urban myth. It may be rare, but it has worked in the past. When experiencing an airline issue, if you ask to speak to management and you articulate yourself in a professional, calm yet confident manner, you can find that you can get a free upgrade (if not this flight, perhaps another for another time) or other perks like free lounge access.
Practicality aside, if all else fails, this is a surefire way of increasing your upgrade odds. Knowing someone working for the airline definitely helps. Even if you can't get an upgrade for free, as family, you can purchase discounted business class tickets.