I think by 'chain' soil' it means the crap that can get flung onto the frame from the chain, especially if you use a 'wet' lube. On a side note, if your chain is flinging lube onto your frame, you can be sure you have too much lube.
A clean chain look like anything else which is clean, however...
Here is what the inside link of a chain looks like when you take it apart (don't do this to a working chain!)...
View attachment 231105
When I clean a chain, the idea is to remove
all of the old lube and whatever grit/dirt has attached itself to it. The best way to do that is with a solvent (disolves grease) and agitation. A brush can't get into the bearings (the circular bits) so I drop my chain into a jar with White (Mineral) Spirits and shake it for a couple of minutes, then rinse the spirits away with something like Muc-Off (the pink soap spray), followed by a shake in a jar of very hot water, followed by a good rinse under a hot tap. I'll repeat this until the water in the jar rinse remains clear. I blow the excess water from the chain and leave it on some tissue paper over a radiator to dry. This leaves me with bare, clean metal for my fresh lube's adhesive to bond to.
Applying new lube should be done as per the manufacturer's instructions, just make sure you choose one which is relevent to your riding conditions (wet/dry).