When I wash the bike, I use a brush and Muc Off (bike cleaner) to clean the chain as I'm doing the rest of the bike.
I dry off the bike and then go to the chain. Using a folded up dishcloth I hold the chain lightly and turn the pedals to remove the wet and muck.
I do this until there's basically no dirt left.
As my bike lives indoors, I then leave it for a few hours to make sure it's dried out properly, and lube it again later.
I use wet lube through the winter and dry lube through the summer*
It seems to work okay for me. I wash the bike after every ride in the winter and probably once a week in the summer.
*summer in Scotland means I probably use more wet lube than dry