Remove the CPU from the socket. Wipe off as much as you can with a paper towel. Then use a Q-tip and rubbing alcohol to remove the rest. Keep using clean Q-tips though and fresh dip of rubbing alcohol. DON'T pour it, just dip the Q-tip in and then wipe the CPU with it. You might have to run a bit hard if your thermal paste is kinda dry. Keep doing it, fresh stuff when the current gets dirty and it'll come off.
Then do a clean one at the very end to shine it up. Then reapply new paste, reinstall CPU into socket, reseat heatsink, and let 'er rip
You can also use isopropyl alcohol to clean off the processor as well. I normally wipe off the main gunk with a paper towel then run a fresh one with some isopropyl alcohol to clean it. Lint free cloth is the best suited for the job but up to you if you want to do so. No need really though...I just use paper towels...Just make sure there are no strands sticking onto the sucker when your done. I examine it carefull after each cleaning since I just use paper towels.
I use 100% cotton balls (not scratchy synthetic balls) and alcohol, then when it is clean, finish with a coffee filter. Coffee filters are lint-free, work great for the last couple of cleanings. Also, 90% alcohol is probably best, though 70% is good and 50% works OK.
I have found that if you have really stubborn paste (dried out), I start with a product found under a few names made by CRC (Brakleen, Lectra Motive). Both products are essentially the same. Use it in the same fashion as the alcohol (Q-tips or cotton balls) and wipe away the paste. It works awesome (no scrubbing). Then I follow it up with the alcohol just to remove any residue left by the CRC cleaner.