I'm a motorcycle camper and I would like to know how you do laundry out on the trail.

Probably do a search for this and come up with all sorts of inventive solutions, one is to put the laundry in a sack (or ziploc baggy, or other water holding container) with the soap and clothes and shake the hell out of it, adding a rock or two may increase the cleanliness. Wring and let dry.

I finally took the time to do a search. I figured laundry was not a popular topic with backpackers but people are indeed inventive. I am considering putting my clothes in a stuff sack, filling it with water and soap, sealing it, then squishing it for a few minutes. Then, I'll wash out the soap somehow. I haven't figured out how to dry it yet as I am constantly on the move.

Would Dr Bronner's work well for this?

Yes, Dr Bronners works for everything, it's like the duct tape of hygiene. Being on the move is a great way to dry clothes in my opinion. Strapping clothes to saddle bags would work well. Just make sure you have them strapped down well... don't want your leopard print undies flying into someones windshield.

I've done laundry in a bucket when car camping. Stopping at a laundromat along the way doesn't take long and does a better job.

In the woods, take some care to keep your suds away from water sources. Don't use a lot of soap. A lightweight all-purpose sink can be made from cutting the top off a gallon milk jug. You can use it for bathing and laundry.The weight is small and with the wide open top, you can fill it with just about anything in your pack, so it takes up little space.

When you have a bear can, you have a washing machine. Post a guard on the food, throw in the clothes, water, and a few drops of dr bronners, and pretend you're a bear after a Clif bar.