No soap on your lock, as you don't want it on the frizzen.
I pull my rifle apart, and I swab the barrel in a bucket of lukewarm, not hot, water.
The lock I will usually spray with the kitchen sink sprayer to help get into places I can't easily get a rag or such, and then I set it under a ceiling fan or by my pellet stove to dry the moisture off it.
Wet patches get run in the barrel to "plunge" out the breech, then dump the stinky water that makes, get a bit more and wet patching till they come out with no black fouling. Dry patches till they come back out dry and no fouling marks.
If I'm planning to shoot soon, I'm done.
If I'm not, then I run a lightly oiled patch down the bore, BUT....have to remember to dry-patch that out before shooting.
I'll work some light oil in under the cock/mainspring to make sure there's no corrosion taking root in there, lightly oil all the parts except for pan and frizzen/vent hole region, and then reassemble.
It's not rocket surgery.