Manuals on Castrametation often cover this topic in addition to the laying out of camp. Other manuals, like The Private Soldier's and Militia Man's Friend are aimed specifically at the regular and how he should keep up such things. The most important thing for a soldier to keep clean throughout the 18th and much of the 19th century is his shirt. It serves as complete undergarment and is the one piece that is boiled and washed with lye soap. This and his stockings are really the only pieces that need to be cleaned constantly, they're also some of the most commonly issued pieces so soldiers should have multiples. This can easily be done in the cooking pot issued to each group or by the many camp followers who function as laundresses. An image from 1782 shows this well. As for the outer garments, they are very often made from wool broadcloth for the fact that it is so durable and easy to care for. If it gets muddy you can let that dry and simple brush it off later. Spot cleaning for stains can be done with water and soap or through simple dry cleaning methods like the use of fullers earth for oil or gin. Leather breeches or trousers were also popular because of their durability and ease in cleaning. They can be washed in soap and water or a white ball rubbed over them to brighten them up.

As for the soldiers themselves, the cleaning of the undergarments does very well to keep the body clean of sweat and oils. Washing hands, face, and other necessary areas was considered standard etiquette at this time and would have been easily done by way of water, soap, and a rag. You won't find references to "bathing" because that is defined by actually taking a submerged bath. Showering today, or basin cleaning, doesn't fall under that definition.