I just use a damp sponge with some non-citrus Dawn for Dishes dish soap, wrap the sponge around the chain and clean away, then rinse the chain with water, rinse out the sponge which due to the soap will pretty much come out looking clean.
I use to use that Park Chain cleaning tool but found the sponge and the Dawn for Dishes to work just as well, plus the sponge and soap are a lot cheaper than that cleaning machine and the solvent needed to use it.