For the paste, 90%+ isopropyl (rubbing) alcohol and coffee filters or microfiber rag you will throw away. Don't soak, just dampen. Works the same on the cpu.
For dusting, I use a paintbrush and vac hose. Remove the fan from the heatsink. Hold the nozzel about 6" away and 'paint' the heatsink. Takes 2 minutes at most. Some ppl use canned air, that works too, just don't hold the nozzel too close or you get condensation buildup, and thats not healthy for the cpu. It also means blowing the dust right back into the pc. Same goes for the fan, just be careful to not let it spin hard as that can damage the bearings and don't hit the blades directly on or they could break with that ice cold air. I usually hit those with the paint brush too. Good for cracks and built up gunk.