As tempting as it may seem, you're not going to want to haul all of that camping gear with you as you hike miles and miles through the mountains. There's the Colorado Trail Trekking Program, which takes away some of the hassle. Hikers who sign up for this week-long service only worry about their daypack, which contains supplies to get them through the hike. Additional camping supplies are waiting for them at the next night's pre-selected campsite.
However, most camping hikers opt to carry their overnight packs with them. Camping is allowed throughout most of the Trail (Waterton Canyon is a notable, if rare, exception). There are a handful of specified campsites, but most campers opt to just go off to the side of the trail and settle for the night. Plus, it's free.
Some things to think about: Be respectful of the small, permanent communities along the Trail, and get to hiking early, so that you can establish camp later on while you still have sunlight and energy.