How do soldiers maintain their personal hygiene while on the front?
There are three body parts that you must keep clean, even on the battlefield. If one of those parts gets infected or causes you any other trouble, the war will be over for you:
Your teeth
I always had a toothbrush with me. I used the plastic bags from the MRE combat rations to protect the brush from dirt. You get used to cleaning your teeth while doing something else at the same time (like watching the terrain, listening to an officer giving orders or checking your rifle). There is really no excuse for not brushing your teeth.
Your butt
You can use wet wipes to wipe your ass. If you don’t have any, try to clean your butt (and your genitals as well) with water from time to time. You should also change your underwear whenever you can. There is always some space in your backpack to take some spare underwear with you.

If there is a break, always dry your feet and change your socks (Photo: US Army)
Your feet
Always carry at least one pair of spare socks with you! Change your socks whenever there is an opportunity and let the old pair dry. Wet wipes or alcohol are perfect for cleaning your feet if you have no water. Sometimes, if the situation allows for it, just take off your boots for a few minutes to let your feet get some fresh air.
You will get quickly used to this “field hygiene”, but sometimes, you will be tired and just want to rest and then you really have to push yourself to make this extra effort and dedicate a minute of your time for your personal hygiene.