Cleaning heat sink grease off a processor is something I hope you don’t have to do often. But if you find yourself needing to remove a processor from a motherboard, it’s a good idea to clean off whatever compound is on the surface of the chip and apply a fresh batch.
You can do it with household tools.
All you need is a lint-free cloth and a good solvent. A lot of people use rubbing alcohol, and others use premium-branded cleaners from the same companies that make the compounds, but I prefer a good-quality lighter fluid like Ronsonol. Ronsonol is almost pure naphtha, and is an incredibly fast and effective degreaser that evaporates quickly while leaving no residue. I use it to remove old lubricants from Lionel and similar toy trains, and it works just as well on modern computer chips. And it’s cheaper than Arctic Clean.
Just squeeze a small amount of the fluid onto the cloth, then rub it over the surface of the chip and the fan. Repeat as needed, but two or three wipes usually is sufficient to clean the surface. The discoloration on the metal will disappear as soon as the naphtha evaporates. If any discoloration remains after a few seconds, give it another wipe. For good measure, when you’re finished, apply a little more of the fluid to a clean portion of the cloth and wipe again. Repeat until you can do it and the cloth remains clean.
Now apply replacement compound if necessary and replace the fan. Less is more when you’re applying heat sink compound–a drop about half the size of a bb is typically sufficient. Heat sink grease’s job is just to fill in the tiny spaces in between the heat spreader on the chip and the fan. It’s a better conductor than air, but a worse conductor than metal. Using a premium heat sink compound like Arctic Silver might lower the temperature a degree or two over ordinary compounds, but the difference is minimal. A proper amount of the cheap stuff will do a better job than too much of the premium stuff.