Timing is a key to finding solitude in the Rocky Region. Go early in the morning - most visitors don't get into the Park until 9 or 10 am. Dinnertime or dusk is also good - most people are "at home" by then. Weekdays are better than weekends. Off-season is better than mid summer.
Hike up the trail a way - most visitors stay fairly close to the trailhead. Choose a less-popular trail. There's a remote Lake Solitude in the Park. Yes, you can even go off-trail into certain areas of the backcountry - check with the Park about these cross-country areas:

East side:

South Cache La Poudre
Mount Dickinson
Cow Creek
Chiquita Creek
Upper Forest Canyon
Lower Forest Canyon
Hayden Gorge
Hunters Creek
Cony Creek

West Side:

Cascade Creek
Hague Creek
Cache La Poudre
Mosquito Creek
Onahu Creek
Nakai Peak
Mount Patterson
North Inlet
Ptarmigan Creek
Upper North Inlet
Ptarmigan Mountain
Mount Enentah
Echo Creek
Columbine Creek