Cleaning the Surface
After you have finished the sanding, and are happy with how smooth the surface is, wipe it down with a clean rag. There are two cleaning and sterilising methods described. Using Isopropyl Alcohol is probably the easiest and safest method, since Isopropyl Alcohol is easy to obtain and it evaporates quickly.
Some guides suggest polishing the surface to make it nice and reflective. Don't do it. Polishing materials are designed for polishing and not for heat transfer. They're nowhere near as conductive as a good thermal compound and the polishing material will inhibit the flow of heat.
Cleaning the surface should be done in a fairly dust free environment.
Don't use the same room that you used to sand the surface.
5.0a Using Isopropyl Alcohol
Dampen a clean bit of gauze with a small amount of distilled water, and wipe it down. This rinsing process should pick up any excess residue from the sanding process. After "rinsing", use high concentrate (90%+) or pure Isopropyl Alcohol to scrub it by wiping it several times. You should only need to drop a tiny amount of alcohol onto the surface. Each time you wipe, use a different section of the gauze or a use new gauze so that you're not putting the same residue that you've removed back onto it.
Repeat the scrubbing process two or three times. Each time you wipe, make sure you're using a clean section of cloth. A good way to do this is to roll the cloth with your finger as you move it across the surface. It may take more than two or three attempts to get it fully clean and that should be apparent if the gauze is a bit dirty after each wipe.
Once it's clean, give it a minute or two to fully dry up before mounting the heatsink to the CPU.
Try not to touch the surface with your fingers when you're done.
Your skin will leave grease on the surface which will also inhibit heat flow.
5.0b Using Phosphoric Acid and Ammonia Water
The idea of using this combination is to combine two fairly aggressive cleaning agents that will both clean the surface well and neutralise each other. The surface may look clean and shiny now, but wait until you see this:
Pour a small amount (maybe 1 or 2 drops) of the phosphoric acid onto the surface. Use several bits of gauze (one at a time) to scrub the surface. They get dirty very quickly, so be liberal with them. As you move the gauze across the surface, roll the gauze with your finger so that you're using clean sections and not smearing dirty stuff back onto the surface. Wipe the surface dry with a dry gauze swab and then put the same amount of ammonia water onto the surface. Scrub the surface, again being liberal with the gauze and again using the same rolling motion with your finger.
Use another dry gauze swab and wipe the surface dry. Repeat the acid cleaning procedure and then repeat the ammonia procedure again. This time you want to be more careful about not wiping the surface with dirty parts of the gauze. Once you have wiped the heatsink dry after the second round of ammonia water, rinse your heatsink with the distilled/de-mineralised water to remove any residual ammonia water and then dry the heatsink with gauze. Make sure you use the same rolling motion with the gauze (you get the idea).
Let it sit for several minutes to dry or if you want overkill, get a hair dryer... Just hope no nits get jet-propelled onto the surface. Once it is dry, it is ready for mounting. The longer you wait, the dirtier it gets.
Try not to touch the surface with your fingers when you're done.
Your skin will leave grease on the surface which will also inhibit heat flow.
6.0 About the Chemicals
Alcohols are best because they evaporate quickly. Isopropyl alcohol is particularly good because it has a fairly neutral pH level. It should be available at chemists or some hardware stores. Be wary of what other agents might be mixed in with it (such as in rubbing alcohol). Some of these agents may not evaporate properly and can inhibit heat transfer. Ethyl alcohol (commonly found in methylated spirits) can be used as a substitute for ispropyl alcohol, but like ispropyl alcohol it should be as pure as possible. Ethyl alcohol may take slightly longer to evaporate than ispropyl alcohol.
Ammonia water is a strong base with the right alkalinity to neutralise phosphoric acid. It is a good cleaning agent and many household cleaners contain ammonia. Ammonia can melt copper, so I like to use it first on copper heatsinks and neutralise it with phosphoric acid. Phosphoric acid is a weak acid (ie does not "dissociate" like sulphuris etc). It is used to halt rust in steel and can be found in many rust prevention solutions. The idea of using a light acid, is that it will help melt away fine burrs left from the sandpaper. As long as phosphoric acid and ammonia water are neutralised and any excess is rinsed off, there should be no adverse effects on the metals or the thermal compound.
Please don't drink ANY of this stuff. It will probably kill you.
Many thanks to mule43, Jester, Blowin' Smoke, my uncle and anyone else who contributed to this guide.