The short answer.
You can camp safely in Colorado during hunting season. The busiest stretch for campers is the fall big game rifle seasons for deer and elk, which typically run in several short seasons from October into November. Wear blaze orange or another bright color when you leave camp, keep dogs close and visible, and expect popular dispersed areas on national forest and BLM land to fill with hunting camps. If you want to avoid hunters entirely, camp in Rocky Mountain National Park or another area where hunting is not allowed.
Colorado Parks and Wildlife sets the season dates each year and publishes them in its big game brochure, so check the current dates for the area you are visiting before you go.