Jumping in a lake is great for freshening up. Streams work as well. No soap required. Hand scrub yourself or use your bandana. Jump in with your clothes and they will be cleaner too. If it's a hot dry day, your clothes will dry while you hike.
You only need to pay close attention in three regions, and two of them are under the arms. Under the arms I clean up with an alcohol soaked cotton pad, which I then use as a fire starter. It's a nice luxury to have all 3 options: alcohol cotton pads, wet wipes, and soap. Choose ethanol over isopropanol, as it is less toxic. I use Dr. Bronner's soap for both backcountry bidets (see video) and laundry.
I prefer zinc oxide based sunscreen in a 1 oz can size. I use one that has the following ingredients: 25% Zinc Oxide. Coconut Oil, Beeswax, Calendula Flowers infused in Jojoba Oil, Vitamin E.
A small umbrella or hat is also a good idea.
Bring along an extra pair of socks and underwear. That's all you'll need. When you get to a stream, if it is too shallow for swimming, you can wet your shirt and wring it out to wash off the sweat.
To wash your underwear and socks, fill a gallon sized zip lock with a few drops of soap (optional 3 drops of bleach) source. Lather within the bag and let sit for 15-30 minutes. Wring out and rinse with clean water.
Dry outside your pack as you hike, or on a rope between trees, or on hot stone. Micro-carabiners make good clothespins. Place clothes on the line, then hook the clip over the line, which is the opposite direction as shown in picture below. Super tiny 0.5in (15mm?) office binder clips also work.
“For smaller items “You can fashion an emergency blanket into a box (reflective side in) and use it to dry wet clothes with either solar or a fire. A small fire is all that's needed and you can dry wet clothes in about the same amount of time as you can with your clothes drier at home (under an hour).” - Bill Stephenson
After sleeping through a rain storm, once the rain has subsided and the sun is out, you can dry your tent by laying it out on hot stone. You can weight your tent down with some rocks, on the side of the mountain facing the sun. You can also hang the tent on a clothesline if there are trees around.
This topic is best covered by watching this video.
How to poop in the woods & perform a backcountry bidet, by Andrew Skurka
Sorry, not an expert on that. I thought the following youtube video was good, but what do I know?