The highway is built jointly by Pakistan and China, and its construction took 20 years and hundreds of lives. Most of the casualties are due to landslides, though somehow, the Pakistani workers took the bulk of the numbers : 810 to 82!! (according to Wikipedia). But when the highway opened, it brought in commerce and better livelihood to the people in northern Pakistan.
Most people starting the journey in Pakistan would either start in Islamabad or Rawalpindi. While there are public buses and minivans thoughout the stretch of the Karakoram Highway, the best transport in terms of flexibility would be a private jeep hire. Northward, the most popular stopover would be Fairy Meadow and Nanga Parbat before reaching Gilgit, the true hub of the Karakoram Highway. Here at Gilgit, you could continue northward along the Karakoram, turn west to Chitral, or turn east to Skardu where K2 beckons. Moving further north, there are plenty of delightful stopovers at various towns, notably Karimabad and Passu. After Passu, you'll reach the border town of Sust, before finally crossing Khunjerab Pass.
There are both buses and jeeps that cross the border to reach Tashkurgan, the border town on the Chinese side. Khunjerab Pass itself is surrounded by snow-capped mountains whole year round and its beauty and scale easily engulfs you. If it doesn't, the altitude sickness probably will (it's at 4730m). Certainly one of the highlights of a journey through the Karakoram Highway.
The "endpoint" of the Karakoram Highway would be Kashgar, Xinjiang China, though there are plenty of people who do the journey the other way round - ie start in Kashgar and down to Gilgit/Islamabad etc. Whichever the way, the journey itself would be the experience of a lifetime....