My spoon, fork, etc are dipped in the pan of boiling water for my coffee after being wiped on my pants/shirt/socks. I heat a tad of hot water in a pan and use my spoon to clean it - no soap is required. If is really dirty, I scrub it our with a handful of sand first. In fact ever since I was a boyscout, I have never used soap to clean anything that I eat off while camping because soap can make you sick and its hard to rinse off. You kind of clean as you go.

I always carry a tooth brush and a small squished tube of tooth paste. I do have a couple CCs of a special liquid soap that I would use for cleaning a wound or some other rare usage. I don't carry soap on short trips, on longer trips a motel soap bar is just right.

There are small travel deodorant anti perspirant things at the drug store. I rarely pack one but often keep one in my truck...

Swimming rinses me off, but you can take a sponge bath in as little as 6 ounces of warm water and a bit of "No Rinse Body Bath". Its the astronaut bath stuff. This stuff really works, removes the stink and you'll feel a lot better. The corner of a small towel dipped in a cup of warm water is nice for cleaning yourself, and then just dry off on the rest of the towel. The best towels are made of microfiber and called "hair towels".

On longer trip personal hygiene and health in general becomes much more critical and time consuming. I'm not much worried about pathogens where I camp, and to that extent I do filter my water, even the water I boiled because it tastes better. I Carry a pur hiker water filter, as its about the fastest and easiest and most reliable that I've owned. Anyway a good supply of clean water is more important to me than washing my hands or god forbid - using soap on my cup or pans.
Jim YMMV
Jim