There is no formal definition of the term 'Expedite' in ATC as far as I remember.
When we use it, we are usually wanting it to help us solve a conflict with another aircraft quicker so that we can then release you on a direct route or turn our attention to another problem. What we really mean is that we want you to achieve a level change with another aircraft as quickly as possible in as short a distance as possible. Therefore what we mean is your best ANGLE of climb. What it often gets confused with (and specified as) is best RATE of climb as that is the only thing we have to measure. (If you have 4000ft to go to get above other aircraft we will calculate how many minutes you might need to achieve that and compare that with your predicted ground position to see whether it will work. Then we predict/ask for a rate of climb to achieve it).
What it should NOT be used for is as the only means of separation, as 'Expedite' to a 757 on its way to Edinburgh will have a different result to an A340-300 on its way to Hong Kong. Radar controllers should always be planning some lateral separation, but the use of Expedite can resolve the conflict quicker and let everyone go on their way sooner. Long expedites (eg 6000ft or more) should be avoided, as this is unfair on the crew and prone to embarrassment as rates of climb drop off as you climb higher.
If it is looking a bit tight, because of the different results of 'Expedite' I prefer to use the following system:
"XXX what rate of climb/descent can you maintain through FL YYY?" I then specify what minimum rate I need and monitor it. Then at least we both know what we expect of each other.
In short, if you are asked to expedite, push the throttles forward/extend the airbrake and gain/lose as much height as quickly as you can over the levels given. Then you will get where you want to go quicker and you might reduce the tightening of some of the sphincters of the more 'gung ho' controllers..
