Go through with your flight as normal, check in, go through security, and find a member of staff for your flight. Ask how much an upgrade costs, and if they have the space available, they're open to offers.
For example, about 5 years ago I got economy+ on British Airways LHR -> LAX for $200, and business offered for $600 or $700 IIRC. Not bad for a 12 hour flight!