Find the hole. 

This one was easy. It is fairly big and obvious. A classic "snake bite" pinch flat. The second hole has already been patched.

Inflating the tube will help locate the puncture site as you can feel or hear the air escaping. With smaller holes you may need to submerge the inflated tube in water and look for a stream of air bubbles.

Mark the hole with the ball point pen. I circle it and add "gunsight"  marks that will still be visible after the cement is applied.

Clean the surface.

The rubber needs to be roughed up and clean for the patch to stick. Buff the surface with the sand paper included in the patch kit. Some kits will have a metal or plastic buffer.

Scuff an area larger than the size of the patch you will be using. Try to get close to the base of any ridges and mold marks.

Sometimes I lightly stretch the tube over the patch kit box to make the buffing easier.

DO NOT touch the buffed surface with your fingers! 

Apply the cement.

A thin layer is better than thick. Apply it evenly and once over an area well larger than the patch. Do not go back over the cement to smooth it out.

At most, squirt some on the tube and spread it with the folded end of the cement tube.

DO NOT use your fingers.

If you missed any spots, let the cement dry completely and apply a second coat.

Let it dry.

The cement must be "dry" for the patch to stick. The surface will look dull when it is ready.

DO NOT blow on the cement or fan it. It will not dry any quicker and the moisture will make the bond weak.

Apply the patch.

Peel the foil backing off of the patch. Center the patch over the hole and press down. This is where the "gunsight" marks are very helpful.

DO NOT touch the back of the patch with your fingers!

Use the rounded end of the plastic tire lever to burnish the patch and firmly adhere it to the tube. Note how the patch has conformed to the molding ridge on the inner tube.

Gently wrapping the tube over the patch kit box can make rubbing down the patch easier.

Remove the clear plastic over the patch and the tube is ready to use. 

Received a tip from Roger-M:

"...in the final step to remove the plastic backing I folded the patch area to crack the plastic then peeled it off from the centre outwards to avoid lifting the outer edge of the patch."