We began our trek at the Kennecott Mines, which is a short van ride from the town of McCarthy, which is (semi) road accessible.  I say semi because to get there you must drive 60 miles on a dirt road and park nearby the town, which is not road accessible, and cross a foot bridge.  Once you've hauled your gear across the foot bridge, you'll be standing at the van pickup site.  

The van will take you to the Kennecott Mines, which is where the de facto park headquarters are.  There is a another HQ in Glennallen which is where you should procure bear canisters if you plan to stay overnight in the park (required).  I'd highly recommend staying overnight, as the glacier, mining ruins, and nearby Donoho Lakes warrant a multi-day trip.

Once we left Kennecott, it's a short 2 mile or so hike to the best camp spot nearby: the area near the Root Glacier access trail.  This is the only one in the park, so use it and prepare for digging a hole the rest of the trip!  While you can pretty much camp anywhere you want in the park, this site offers a step above the rest for it's easy glacier access, water availability (although you're right next to a glacier, it is receding and it's somewhat of a cliff much of the way), and stunning views.  Note that porcupines are numerous around here - so watch your food or a spiky bandit may make off with it (this happened to us)!

Set up camp and hike the 14.4 mile out and back.  An alternate route would have you camping roughly 7 miles up the trail near the Erie Mine spur trail.  This is what we ended up doing since we got a late start to the day, however I would caution against it unless you have done this trail before and know exactly where this is.  We didn't end up finding the correct campsite until we were on our way back, as it is fairly difficult to find AND there is no water nearby.  Also to note that the Erie Mine is an extremely difficult route (we didn't attempt it) and people have gotten stranded up there as the trail is much more difficult on the way down.

The closer you get the the Stairway Icefall the more stunning it will become.  We hiked up to the Knoll for a perfect view.  One of the most memorable hikes of my life!