Trekking through the mountains will make you dirty. You will sweat as you haul your equipment up to take-off. You will sweat as you hurry down the mountain, in order to escape an approaching thunderstorm that you could not see when going up, because it was at the other side of the mountain. Flora and fauna will stick to your body. Especially when ambushed by bushes and trees after loosing your way. They will leave all kinds of traces on you and your clothing. Mud, sand, seeds, leaves, and twigs will dress you in combat camouflage that makes even the toughest marine green with envy. Animal excrements or (now dead) slugs will colour your butt because you forgot to look before you sat down. Spider webs will turn you into a walking mummy when you are the first one to use a trail.
This all sounds awful, but dirty does not necessarily mean bad. Trekking for a few days or even weeks without washing yourself has the advantage of creating a (maybe too) large privacy sphere around you. This comes in very handy in crowded places such as long queues and overloaded trains. It might create some annoying or surprising looks though, and possibly evoke actions that you are not mentally prepared for. For instance, people might spontaneously start throwing buckets of water at you (has been observed at Place de Verdun, St. André-les-Alpes), order the largest bottle of Perrier available and empty it over your head (has happened to me on the Chalvet take-off with a bottle of a less expensive brand on the first(!) day of a vol bivouac), or kindly but firmly invite you for a swim in the nearest stream or fountain. The most clever solution would probably be pouring a bottle or glass of lemonade over you. This seems illogical, since it will make you even more dirty than you already are. But the sweet sticky liquid is likely to attract a swarm of wasps and similar aggressive insects that you can only get rid off by jumping in the nearest bath available.
If you trek with a group, they will usually throw you in a stream or a lake before the privacy sphere gets too big. If you trek alone, then try to do the same on a voluntary basis. It is harder to notice the smell when you are on your own, but you can safely assume that your bath is overdue when your clothing is sticking to your skin and creates a Velcro like sound when taken off.
In summer you will generally sweat (a lot) more than in other seasons, making you (feel) dirty sooner than hoped for. Luckily, summer is also the warmest time of the year. So, a bath every day is a pleasure rather than a survival challenge resembling some strange old Scandinavian tradition where people comfortably relax in a frozen lake, after hacking their way through a few feet of ice in order to access the almost freezing water below.
I take a slip, short sleeved shirt, and a pair of socks along as spares. This allows me to wash and dry the dirty ones, while I wear the others. The other items (my trousers and long sleeved shirt), do not get that smelly quickly and get washed less often. Preferably on a resting day with sunny weather and abundant time to watch all items dry. This approach allows me to have relatively clean clothing (and body), while it minimizes the (extra) weight I have to carry.
Please do not use soap when washing or bathing in a stream or lake! Nature has great difficulty in processing soap remains. There exist environmentally friendly soaps, but I have not tried those yet. If you are really dirty, a bit of sand will work fine. Soap just adds weight that I can do without. Besides that, you do not want to pollute a stream (that is possibly used downstream for other purposes such as drinking water) with (environmentally friendly) soap. The main soiling factor for your clothing is transpiration, followed at a distance by dirt. Water will get rid of the transpiration and most of the dirt. The dirt that does not get out and requires soap, gets washed out after your trip.