Generally speaking, a "hobo shower" is a method of bathing that can performed in a public restroom. There are quite a few methods of doing this (each independently devised millions of times across the globe), and in practice they aren't restricted to the homeless.
There are two main types of hobo showers - the mild and the thorough. The former is what usually happens when you get back sweaty from the gym or haven't had time to take a shower in the morning; a hoopty five-minute fix for those who have noticed themselves smelling a bit foul. The latter, however, is considerably more involved, and might take as long as fifteen minutes - it requires a good continuous source of hot water and soap, which usually precludes the latrines at Denny's.
The first step to taking a hobo shower is finding a cleansing agent. Usually it's hand soap. But for those who find themselves in inadequately-maintained restrooms, there are plenty of alternatives. Moist towelettes, shampoo, rubbing alcohol and spray disinfectant (in roughly-descending order of hazard) are just a few of the materials easily adapted for hygiene. Be creative. But remember that you're trying to do two things here: get the smell off and keep it off. Rubbing potpourri-covered fingers on your sweaty 'pits without doing anything else will make you smell like a sweaty hobo with potpourri on your armpits. Likewise, getting some hand soap on the underarms can get rid of the sweat and bacteria there but in a couple hours you're going to smell almost as funky as you did before. However, you're not likely to have anything that can do both of these things. Although hot water never hurts. Rinsing completely after a hobo shower is a bit counter-productive, as it will remove the soap intended to mask your stench. However, the first round of soap you use is going to stop smelling like soap and start smelling rank, so you may want to rinse that off. But after you do that, apply another round to keep you fresh throughout the day. Dab (don't wipe) it dry with a paper towel. Nothing's worse than having massive stains on your shirt. And while you're at it, you're probably going to lean over the counter (assuming there is one) a lot. Wipe it off with a towel first; from my experience bathroom counters are precisely at crotch level and covered with water from whoever went before you, so if you lean over without wiping it off you're going to look like you whizzed yourself for the next half-hour.
Next, you'll want to make sure you've gotten everything covered. The armpits are probably the most egregious offenders, but your nether regions can raise a stink too. Especially if you haven't showered in a couple days, and especially if you've dropped a deuce and been unable to wipe thoroughly. You might not think your butt smells weird. It does, you just can't smell it because your nose is on the other side of your body. Even if you could turn your head far enough to smell it, you shouldn't use your nose to judge your handiwork anyway - people don't notice their own smell because they're used to it. This is why the guy who sat next to you at the diner didn't pass out the next time he dropped trou. Rather, judge your smell by the amount of scrubbing you've done. Don't bother holding your hand up to your nose, either, because it will only give you a false sense of security.
For a thorough hobo shower, the process is more stringent. Instead of sticking your hand up the side of your shirt, you may need to actually take said shirt off, and even for a guy this will arouse the suspicions of anyone in your vicinity. Scrubbing yourself in a public restroom is going to make you look like a hobo. There is no way to avoid this. However, if you do it right, you won't look (or smell) like one by the time you leave. Keep in mind that, for the most part, you're only doing damage control. You're putting off the onset of a stench until you get home and have the opportunity to clean yourself fully. Even a thorough hobo shower is a losing proposition, because shampooing your hair is more or less impossible and the amount of water you're using doesn't remove the dirt as much as neutralize it and spread it around.
Hopefully, if we all do this once in a while, the world will be a less smelly place.
This was written under the influence of my medication, whose side effects apparently include making your writing suck. I will revise it when my prescription runs out in a couple days.