Just very, very occasionally airlines will offer deals on upgrades but I would never rely on it. As 1DC says the best way to get a good deal is book well in advance (as much as 90 days) and be flexible. If an airline is going to offer an upgrade at checkin they will normally do it according to a set of rules that don't include allowing someone to bowl up and shell out some (small amount of) money - it has happened, it does happen but it's rare; when it happens it often means the airline has messed up yield control in some way.
Unfortunately, trying to upgrade a ticket between purchase and departure is likely to prove surprisingly expensive, phone reservations and ask but make sure you're sitting down when they tell you; it can be that the upgrade (not the total of previous fare + upgrade) can cost MORE than the ticket in the higher class would have done had it been purchased in the first place.