The area you're talking about falls entirely within the Mesa Ranger District of the Tonto National Forest.
I haven't ever set up camp that close to town, but there are numerous recreation sites and campgrounds near the river. Virtually all of them require the purchase of the Tonto Pass.
A map of the forest (click #3), and links to all the campgrounds and recreation sites can be found on their website
Because it is so close to metro-Phoenix, most areas along the lower-Salt do not allow camping outside a developed campground to help protect the fragile environment from too many drunken yahoos.
You can get more information and buy much more detailed maps of the area from the Mesa Ranger District