The phraseology "expedite climb" doesn't mean anything. Nobody here has been able to give an official response with ICAO definitions. Why? Cos there ain't any, thats why.
stands by to be proven wrong
What it does do, I would hope, is convey a sense of urgency to the pilot, ie: whatever you're doing, do more. To use "expedite" means the controller has made a mistake...or maybe the pilot has made an error, but a conflict has been spotted and its a quick easy way to let the pilot know something needs to be done and it needs to be done now, no questions. It should include some traffic information as well, which should give an idea of exactly what is expected. If an exact or minimum climb rate was required it should be given, ie: "climb with 3000fpm minimum due opposite traffic" but " expedite climb due traffic" takes less time to say.
Basically, if all is well, the word expedite will never be heard, but if, as a pilot, you hear it, you can be sure something has gone wrong, and any reaction is better than none, the more exaggerated the reaction the sooner the situation is resolved.
I am talking in my capacity as an Upper airspace controller in Maastricht, I know that different situations may arise in Approach or other environments which might lead to the word being used under totally different circumstances.