The way that some people bathe on the Appalachian Trail is when they stay in motels or hostels along the way of the Appalachian Trail.
Some people also just do something like a sponge bath and may use water from a creek to bathe and stay clean.
You can stay pretty clean by just doing a quick sponge bath to wash away some of the sweat or even yourself a solar shower that is for camping and hiking.
You fill the solar shower with water and the heat from the sun heats the water up so that you have a nice warm shower when ready.
Some people camp in tents along the way and some people stay in some hostels or hotels in towns along the way of the Appalachian Trail.