Take off the nut holding the clipper blades together with an adjustable wrench or pair of pliers. This will make the blades easier and safer to handle and sharpen. Wipe away any grime from the nut, bolt and blades.
Set a mill file at a roughly 25-degree angle to the clipper blade, or to match the bevel, whichever is easiest for you to do. Slide the file down and across the face of the blade a dozen times.
Turn the blade over, set the file again and slide it down and across the other side about a half-dozen times to remove burrs from the blade.
Repeat Steps 2 and 3 on the other blade.
Apply a few drops of oil each to the bolt and both blades, and rub that in with an old rag.
Reassemble the clipper blades and tighten the nut back onto the bolt.