I'm just going to stick to the laundry question on here and ignore all the other banter.
Some tips to keep in mind:
Anti-bacterial soft soap will work sufficient, you can always acquire some from a public restroom. If it's in a big dispenser, then carry a tiny bottle on you to fill up. You'll need some anyway to bathe every now and then(if you prefer to do so).
Use whatever you can scrounge up and is clean enough to wash them in, discarded bucket, empty old cooler, the water container of a dehumidifier, baby pool, whatever you find on the trash. Since I always carry a few construction strength trash bags, I sometimes dig a small hole in the ground and line it with one. Do your laundry that way.
You can do it without a washboard, just put some elbow grease into it. There are many good reasons to wash your clothes every now and then. Lingering residue from past spilled meals attracting animals, is one that comes to mind.
Also growing bacteria from said stains. Also if you hitchhike instead of hopping trains, I'd consider cleaning up every now and then. I generally just worry about keeping my socks and underwear clean for hygiene reasons, unless my outer clothes start to smell too bad. Those I can wash in a large plastic shopping bag.
Dry them on a line or cut some saplings as poles to hang your clothes on.
Above all, just be resourceful and improvise. That's what I do. If you can't find a way, make one!