Wasatch Mountain State Park Campground
Wasatch State Park Campground is just over the hill from Park City and is your best bet for camping near trails with decent access to Park City. From the campground, the WOW trail and Dutch Hollow trails are ridable from camp, and you're only about a 15 minute drive to the top of Guardsman Pass where you can drop into Deer Valley or hit the Wasatch Crest.
Spruces Campground - Big Cottonwood Canyons
Spruces Campground is located in Big Cottonwood and is about a 30 minute drive, albeit a beautiful one, from Park City. From here, you could easily ride up the road and hit the Wasatch Crest to Mill D trail that would finish near camp. Otherwise, you'll be driving to Park City for additional ride options. Note that Big Cottonwood is part of SLC watershed and does not allow dogs.
Jordanelle State park Campground
Jordanele State Park offers traditional campground accommodations and is a quick ten minute drive away from Park City. From camp you are a short ride away from Spin Cycle & Village Trails, and could easily loop into the Deer Valley trail system from here. Note that Jordanelle State Park & campground is very exposed and does not offer much in the way of protection from the sun during the hotter summer months.
River's Edge Campground & RV Park
River's Edge is a private campground and RV resort with great access to the Middle Provo River. While it's geared towards the flyfishing crowd, it's a good option for those traveling in an RV or travel trailer that's not too far away fro Park City. Coyote Canyon and Dutch Hollow trails would be accessible from camp via the rode and Park City is a short 15 minute drive away.
Uinta National Forest Camping
The most primitive and remote camping options would be located off the Mirror Lake Highway in the Uinta National Forest. While there are a plethora of trails located in the Uinta's few of them are optimal for biking and are best suited for hiking or gnarly adventure style riding. While the Uinta's likely offer the best true camping experience, it could easily be an hour+ drive to singletrack from camp. Note that the Uinta's are prone to bugs in the early summer months.