This is how I do it:
You need an alcohol-based cleaner and some cotton buds. I use AM record cleaner as it's designed to clean with no residue.
This is all at your own risk! It takes time so be patient.
Get the victim - er, patient - and position the disk such that the offset hole on the spindle is at the top- so you know where to stop

Spray the cleaner onto one end of the cotton bud. Now hold the disk open by pulling the metal guard.
Wipe the wet end of the bud on the disk surface such that the visible surface changes colour. Don't be afraid to touch the surface, but don't be too heavy. Just enough pressure to rub off dirt and oxide.
When it's covered, use the dry end to dry it again. It'll still be wet, but you need to get the excess off otherwise it gets into the fibre inside the disk.
Turn the disk to expose the next dry segment of the disk, and repeat. If the disk doesn't get wet, you need a bit more cleaner on the bud.
Keep turning the disk - don't let it stay in one place too long until it's dry.
When the offset spindle hole is back to the top again, you know you've done the disk.
At this point keep the cover open and keep turning the disk until it's at least mostly dry. When it's dry it'll go back to it's original colour.
Then try it

Remember this is just what I do, your mileage may vary!
I was thinking of making an instructional video, would that help?