I am going to show you how to clean a chain while its on the bike, using this Park Cyclone Chain Scrubber. You put fluid in here; there is a fill line, then you put that chain cleaner up against the chain. Turn the crank backward. I think Park suggests turning this 30 times, and you can see the chain is considerably cleaner, and the fluid has gotten quite black.

You can just go ahead and do it a second time. After you have run that chain through the chain cleaner once or twice, go ahead and wipe it. If you now put lube on there, the oil will just float to the top of the water, it won't really get down into the chain anyway, so really the chain needs to dry before the lube will do anything. If you have to go riding right away, at least put some lube on there. So that's all it is to cleaning a chain with a Park Cyclone Chain Scrubber or a similar chain cleaning tool like this.