I've been out there a couple times. Both times I've camped about 15 minutes outside of Lake City, just at the start of the Nellie Creek Rd. (That's the road that goes up to the Uncompahgre Pk. trailhead.) There's about four sites up there. It's National Forest, but not an established site, so it's free. Those sites are also off the main loop, so you won't be right in the dust of the vehicles on the road. There's a vault toilet about 200 yards away on the main loop.
Also, I would recommend you do the Alpine Loop in a clockwise fashion. That way you will be on the inside of the road when you're on the most exposed part of Engineer Pass.
You can do the loop in one day. If you want to stop a lot and look around at the old mines and ghost towns, make sure you get an early start, or set aside an extra day.
I also found that the portion of the loop that goes into Ouray was the roughest overall section. My opinion that the primitive at-large camping near Lake City is more available then near Ouray, so Lake City is the place to start. Plus Lake City is closer to Colorado Springs.
If you have any other questions, just ask. I love the San Juans.