You are probably looking forward to following a water truck naked for a mobile shower based on some old videos you've seen. This is a truly burny solution for getting clean, but please don't. I know this looks like fun and you are dying to try it, but the water is not potable and some burners got sick from it a few years back. Supposedly the water comes from a cattle pond and has high e coli and coliform bacteria (not sure I spelled these correctly) counts. This is probably the only thing that really disappointed me at my first burn.
There are a variety of safer ways to get clean at the burn and I'll discuss some of them here. Bear in mind that it is impossible to stay clean for long anyway.
Moist wipes (baby wipe type) are pretty much an essential in a place where water is scarce and all waste water needs to be carted away when you leave. These should be the unscented type since the scented ones can be offensive after a while. A wipe bath will get you presentably clean and they are also useful for washing your hands. Should you have some intestinal distress, they will be a blessing for post potty clean up as well (carry a zip lock - these must not be thrown into the porta potties).
Many wipes in their original packaging are prone to drying out. Repackaging them into smaller quantities in zip lock bags is highly recommended. It is also handy to carry some in a zip lock when your are out and about.
I have experimented with various brands and so far find that the wipes sold at Costco are the best value - least likely to dry out, large size, and low cost. Get the regular ones, not the "flushable" kind since they are larger and more durable. The downside is they only sell them by the case, so you may want to split a case with a few other people to defray the cost.
Unlike normal camping where water disposal is not an issue, camp showers are a logistical nightmare at Burning Man. You must use a catch basin such as a kiddie pool or something you've built. A strategy for evaporating as much water as possible is necessary as well as a plan for getting the water that doesn't evaporate into buckets or drums for removal from the playa. The shower itself may be as simple as a solar shower bag and can be wide open to fully enclosed depending on your modesty level.
Because you have a limited supply of water and even more importantly need to cart away the gray water, conserving water is smart. Wet yourself, soap up, then rinse. The long luxurious showers you may like at home just have no place on the playa. Even if you have an RV with gray water tank, strict water discipline must be observed or your tank will be filled in no time.
RV Tip - To prevent accidental overuse of water (someone who isn't aware or water valve not completely shut), only turn on the water pump when you are actively using water and shut the pump immediately afterwards.
No Rinse Shampoo
Most sporting goods stores sell no rinse shampoo in their camping section. This stuff was supposedly developed for the space program, and may get your hair marginally cleaner. I am not impressed by it, but if you don't have shower facilities it is better than nothing.
Since I am trying to make this guide timeless, I don't want to be too specific as camps come and go over the years. If you check the WWW (the schedule book they give you at the greeters), there may be camps you can seek out for some degree of cleaning. The Human Carcass Wash comes to mind - basically you strip down and walk through while being scrubbed by lots of human hands while also taking your turn to do the scrubbing. Fun and getting cleaner at the same time.
Embrace The Dust
Regardless of how much effort you put into getting clean, you won't stay that way for long. Be mentally prepared for a level of dust and filth that you would not be comfortable with in the default world - there is just no way to avoid it.