Stand at the bottom of your cutter.
Look at the bottom of the rod that holds the cutter head. There should be two allen screws. If you remove those screws, the thing-a-ma-bob that is holdiing the rod in place will loosen and you will be able to remove it. Then the cutter head will slide right off. I use lighter fluid to clean everything really well. Before putting it back together, I usually do a last cleaning by pouring some fluid on a piece of matboard and using circular motion run the cutting head through that several times. It always seems to get extra dirt off. I do it until nothing more comes off.
Hope this helps. It should be pretty simple once you start to take it apart.
Edited to add: I really do love my C&H cutter. I have a really nice Fletcher at work, but when I come home and work on my C&H I am reminded how nice those cutters really are.
Candy