Along with the thrills that come with battling the river, you'll also have to sacrifice many of the conveniences of home. One of these is trash pickup. The NPS doesn't have any kind of trash pickup for their campgrounds. So the courteous thing to do is to have the least impact and leave as little evidence of yourself as possible. This takes some planning.
You must remove all your waste -- including organic waste. This means that you can't even leave behind human waste or ashes from a campfire. Food waste attracts rodents and ants. And these ants will do more than invade your crumbs and food supply -- they will also bite.
Luckily, the NPS has some helpful recommendations for good camping techniques. For instance, they say you should ideally start camping at a location during low tide. You can do your activities (such as bathing and dishwashing) on the beach at low tide so that once the high tide comes, it can help clean up for you, leaving a pristine beach behind. The NPS provides handy tables for you to figure out which locations will have low water at what times.
If you are committed to making as little impact as possible, the rangers at Grand Canyon suggest that you choose a camping site with willows and Tamarisk. These plants indicate that it is the post dam area flood zone. Avoid areas with native plants like Mesquite trees, which indicate the old high water zone and take a long time to recover. If you do find waste leftover from previous campers at your site, consider taking it with your own trash.