How do I grasp an upgrade from economy/coach to business/fist class?

flying from DFW to Edinburgh.

Answers: Buy a first class ticket, or pay the difference contained by fare. If you're a frequent flier with the specific airline you start to be flying to Edinburgh on, cash within some miles and treat yourself to first class.
No upgrades into buisness or first class on International flights...sorry.

by paying for it. Airlines no longer upgrade for free even if they screw up