Just because you're backpacking, that's no reason to stumble around like a sweaty mess.
This post is mainly for ladies who want to look pretty while they travel, but men, please read the first 3 sections as you need to be presentable too. And not smell.
If you're somewhere far from showers or baths, you still need to wash the important bits daily so you don't get ill or offensive. These parts are:
If you're struggling to find washing facilities, have a Glasgow shower - just spray on some deodorant.
It's also good to pack talc for your feet and other bits that sweat a lot. Using talc can make you feel more comfortable as well as absorbing sweat.
If you know you're going to have trouble washing your hair as often as you would like, pack some dry shampoo. It can add days between washes! Talc will work as dry shampoo in a pinch.
Also pack a hat, bandanna or buff to cover your hair when it's disgusting.
Shampoo bars take up less space and are lighter than normal shampoo, but they don't work quite as well.
Avoid packing anything that creases easily if you can help it. If your clothes are creased but you don't have an iron, hang them up in the bathroom (especially if there's hot water) and the steam will help the creases drop out. Or, put your clothes on a bit earlier than you need to as wearing them will help the creases drop out.
Roll, don't fold, to minimise creasing.
If there's no laundry available, use concentrated travel wash in the sink to clean your clothes. If you haven't got any, use any cleaning product except soap (shampoo, shower gel etc). You can also get in the shower with all your clothes on to clean them quickly.
To dry clothes out, lay them out on a towel, then roll up the towel with the clothes in it. Twist both ends as much as you can (it's easier if you have a friend holding one end). Then hang clothes up in the sunlight if you can, over furniture if you haven't packed a travel washing line.
Stain remover wipes are great for travel, especially if you're a mucky pup.
Use combination products whenever you can (eg foundation and powder in one, or eyeshadow that can double as liner when wet).
Make sure you clean your applicators periodically.
Take little products (eg those trial mascaras you get in those freebie bags) - not just to save space but to make sure you use them up before they dry out. On the road, people tend not to wear as much make-up as they do at home.
Your lipstick will go all melty and horrible. Pack it in a separate plastic bag, take a spare, and/or use a palette and brush. Tubes of gloss work well as an alternative.
If going to the tropics, pack make-up for hot weather, so it doesn't all slide off your face. For example, a BB cream or tinted moisturiser is a better bet in the heat than foundation. Go light on the eye make-up too.
Be very careful about using nail salons abroad - bleeding cuticles can open the door to infection if the equipment isn't properly sterilised. It's best to do your own (or leave them unadorned) if in the developing world.
Clear or pale polish won't show chips as much.
Use nail polish remover pads instead of a bottle of remover. If that stuff leaks in your bag, you're jiggered.
Lots of places offer cheap massages and foot massages which are a great way to relax. Choose your massage place with care though - if there are no female clientele it could be the wrong sort of massage parlour.
Don't get anything pierced or tattooed in a country with lots of HIV positive people (or high rates of hepatitis), nor anywhere that hygiene is poor. It's not worth the risk. If you want something to remind you of your travels, you can get it done at home.
If there's a current thing for getting your hair done in a particular way (remember wraps?) go ahead and do it. It will look dated when you look at your pictures in a few years, but it will make you feel all cool and bohemian at the time.
Buy some nice jewellery. What a great momento of your trip it will be!
If you think you can travel and stay clean at the same time, why not have a look at some of the wonderful places you can go on a career break?