I usually pack enough clothes to last a whole week before it is time to wash them.
At the end of each day the dirty clothes go into a plastic trash bag that is then packed into an outer bag for protection from the elements. This same outer bag also contains the sleeping bag.
Laundromats sell small boxes of detergent so I don't carry any on the bike.
There isn't much more to say on this topic. The clothes get washed, dried, repacked, and worn again.