The rock cut you're about to drive through was blasted out in the early 1900's when the railroad to the copper mines of Kennicott was being built. The railroad began in Cordova and followed the Copper River to current day Chitina before turning through the rock cut and heading east towards the Wrangell Mountains. In the 1960's the railroad handed over the land, and liability, to the newly established State of Alaska which promptly began pulling up tracks and creating the road which facilitated the rejuvenation of McCarthy and Kennicott.
You should count on a minimum of two hours to travel the 60 miles to McCarthy. Take your time. You are about to travel through incredible, dramatic, pristine wilderness that is part of the McCarthy/Kennicott experience, and for many people a highlight of their Alaskan experience. You are driving into the heart of America's largest national park, with a good chance to see swans, moose, grizzly bears, black bears, owls, eagles, lynx, and other animals that make this area home.
There will be many times when you feel the need to stop and take a picture…do it! It may be an abundance of wildflowers that catches your eye, a gorgeous lake with a family of ducks, or a view of mighty mountains in the distance that takes your breath away. This isn't Denali, where you need to park at the entrance and ride a bus—it's an incomparable wilderness experience with no such restrictions.
Regarding road conditions, you may hear horror stories about the McCarthy Road. Since 2005, the State of Alaska has spent millions of dollars improving road conditions that make these stories, and the concerns they raise, less and less relevant. Stories of tough conditions are not reasons to skip the drive or take a flight as much as a heads up to take your time. It is recommended you travel with a spare tire, jack, and some snacks and drinking water.
If you're in a rush, beware. Conditions vary depending upon volume of traffic, weather, and the last time the state sent out the Grader machine. In addition, the significant road improvements of the last 10 years have also facilitated speed. There are more tour company-operated, 15-passenger vans (driven by 20-something guides) lurking around corners. Also lurking around the bend may be a large pothole, frost-heave, or other aberration that can quickly eat up your car and provide you with your very own horror story.