Interesting question - and one whose answer will vary GREATLY depending on where you are!

We camped nearly exclusively throughout Canada and the USA. In big cities we stayed with warmshowers hosts, but otherwise we were camping out. Sometimes we managed to find campgrounds with showers - LOVED it when that happened! Other times we bathed in lakes or rivers. Other times we simply didn't bathe.

As for clothes, we typically wear the same clothes until we can wash them. We may wash them in a river, but we could generally find a laundromat at least once a week in some small town we were pasing through. We stopped there with our bikes and changed clothes in the bathroom before throwing the dirty ones in the washer. It worked! the one exception is my cycling shorts - I hate wearing them more than one day! I carry four pairs and have never (in 3 years on the road) had to wear the same pair twice. I wash them out wherever I can find and strap them on my rear rack to dry.

Once we hit Central America, things became more difficult in the bathing arena - there weren't a lot of lakes and rivers and those we did find were freqently filthy. Then we started checking into hotels more often - many time specifically in order to find a shower!