'Double-clutching' is a term from car racing. Race cars don't always have synchros because a race car engineer will see a synchro as non-essential to the car's ability to go fast. A synchro is therefore 'useless extra weight' on a race car and will be thrown into the trash faster than you can say "I stuffed a car on the corkscrew". Double-clutching in race cars without sequential transmissions is therefore a requirement to drive them....here's why:
As far as a transmission is concerned, synchros are really there more as a convenience than anything else. A synchro's function is to help the dog teeth on the drive collar mate up with gear the driver wants to select. They help the engine revs match up to the new gear ratio being selected. The effect is hard to detect when you're driving because while the clutch is depressed during the gear change you won't feel the impact of the synchro through the engine. Inside the transmission though, during the shift, a synchro is hard at work accelerating or decelerating the gears and layshaft in your transmission so that the dogteeth on the drive collar can lock into the newly selected gear.
A well trained driver doesn't require such niceties to get a car through the gears. Any driver worth their salt will have the gear ratios of their car etched on their mind and will automatically use the throttle to get the revs where they need to be for the next shift. It's not hard to do, and if you think about it a bit, you'll find yourself doing the same.
On a production car, using the throttle to match the upcoming ratio is easiest to do on a downshift. Imagine yourself slowing down in your car. You need to come into 3rd gear from 4th.
A normal driver will do the following:
-Depress the clutch
-move the gear shift from 4th gear to 3rd
-release the clutch.
A good driver will do the following:
-Depress the clutch
-move the gear shift from 4th gear to neutral
-release the clutch and blip the throttle to bring the engine speed up to the new ratio while making the gear change to 3rd
-depress the clutch
-move the gear shifter from neutral to 3rd
-release the clutch
A race driver will do all that a good driver does, but will do it while also heel-toeing on the brake (applicable only to clutched cars), looking in their mirrors, and in the case of cars that stop really quickly, they'll be mostly blind during the process (Formula 1 cars stop so quickly that the driver's eyes deform from the G-forces and it temporarily impairs their sight).
This is why it's called 'double-clutching' because the clutch is depressed twice during the gear change. In actual fact, you don't have to depress the clutch twice, it just helps the engine AND the transmission come to the new revs required to match up the drive collar. Without the double clutch press the engine speed can be matched, but the internals of the transmission will still be spinning at their leisure. Race car mechanics will also point out that a single clutch press leads to parasitic loss in the transmission because the engine's power is being wasted for microseconds and isn't translating into motion where it is needed most, but that's splitting hairs.
In a production car, double clutching on an upshift is harder to do because production cars are universally equipped with very heavy flywheels which make the engine revs fall slowly when the throttle is released. This makes it hard to make a fast upshift using double clutching because you have to wait around while the revs fall. Contrast this to a race car which typically will have a very light flywheel; when the throttle is released, the engine revs dive FAST.
In your case, to make the change to 5th gear, you will depress the clutch, move the gear shift from 4th to neutral, release the clutch, coast for about a second to let the engine revs fall, depress the clutch, move the gear shift from neutral to 5th, release the clutch. If you've judged the target revs correctly, the synchros won't have been needed to complete the shift smoothly and you won't get any buzzing sounds from the dog teeth as they match up. Trust me, if you practice this for a little while your shifts will be as smooth as butter.
As an added bonus, driving in this manner is better for your engine, better for your clutch, better for your dog teeth and better for your synchros. This is because if you are matching engine revs to the gear ratios then the synchros never have to eat up any unnecessary forces that the wheels chuck into the transmission when gears are changing and the engine won't have to get spun up aggressively by the wheels when the clutch is released.
But don't listen to me, let's watch the master himself at work. Senna uses double-clutching several times (though not exclusively) both on up and downshifts in the following video.
http://www.youtube.com/watch?v=8By2AEsGAhU
Don't know who Senna is? You must:
http://www.youtube.com/watch?v=aNmqn3heGgE
I hope this helped. It's a tad hard to explain, so if I've only confused you more, ask some follow-up questions.