Now that you have the heatsink, or heatsinks off, it's time to clean the old thermal past off in preparation for applying the new. I use a few drops of Arctic Sliver's special removal compound to soften the old past and make it easier to remove. Use a lint-free cloth or specialized electronics wipes for this, NOT paper towels. These can scratch the smooth surface of the heat spreaders on the GPU and CPU.

Once the old paste is gone from both the chips and the surface of the heatsinks that go over the chips, do a final pass with an alcohol wipe to remove any lingering traces of thermal paste and fingerprints. The chips themselves should look almost like mirrors.

Don't worry about it is there is some residual paste around the edges of the chips, as seen in the second image. The surface is what we're worried about here.

When applying the new thermal paste, make sure to read any instructions on the packaging. Not all thermal pastes are the same. For this job, I used IC Diamond, which requires pre-heating in a cup of hot water. Also, when applying it, use a portion about as large as a pea. It's better to use a little more than too little. Excess won't hurt anything.

When you've got the thermal paste applied, place the heatsinks back on their respective chips, taking care to place them directly on the thermal paste, and using a back-and-forth rotating motion to evently spread the paste. Then secure them with the screws, following the numbered order on the retention clips.

You're all done! Now reassemble the notebook, and fire it up. There should be a noticeable difference in fan speeds and temperature after the repasting.