In case the processor and/or cooler have thermal on them, you should remove it before applying a new thermal grease.
In Figure 8 you see the case of a Sempron processor that was originally mounted with a cooler that already had thermal compound.

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Figure 8: Sempron processor with remains of thermal compound.
In this case you first have to clean the rests of thermal compound. Use a cotton swab with isopropyl alcohol (isopropanol). This kind of alcohol is purer (doesn’t contain water) and can be found at electronics equipments stores or drugstores.

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Figure 9: Clean all the remains of thermal compound.

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Figure 10: The processor is clean and ready to receive the thermal grease.
You can also clean the rests of thermal compound that remained on the heatsink. We used an utility knife to rub out the rests of the compound, being careful not to scratch the cooler base. To finish, use a piece of cloth with isopropyl alcohol (isopropanol) to remove the rests of thermal compound.

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Figure 11: Cooler with rests of thermal compound.

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Figure 12: With a utility knife, rub out the rests of compound.

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Figure 13: Final cleaning.

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Figure 14: Cooler ready to be installed.
With all remains of thermal compound removed, you can now apply the new thermal grease.