I did it in 2005. I stayed near Rennes les Chateau and rode all around that part and France for a couple weeks. A couple days down into the Pyrenees and up the Port de Pailheres, one of which was to see the Tour go over. Access was the least of my concerns on a bike. Issues you have are adequate water supplies, food, protection from the heat, the climb itself. Its easily my favorite day on a bike, and one of my favorite days in my life.
In a car its an entirely different matter. Access will be an issue. You'll need to drive up there at least a day in advance to get a decent spot of your choice, and you may need two or more if its a finishing mountain, one of the greats, or one predicted to be a pivotal climb.
As others have said, find out ahead of time when certain routes close. The day I intersected the Tour, the Pailheres was not the finish. It continued onwards to Ax les Thermes. Cars were backed up on the lower slopes some 3-4 hours before the peloton came through, while I was riding up. There was no traffic at all on the upper slopes and I suspect the road up there had been closed a lot longer. There were of course many cyclists riding up until the very last minute.
Get your accommodations very early. Start looking now. Book it soon. If you are riding up, you've got a few logistical challenges like getting your bike in there to your place of stay. I strongly suggest a hard case. Pack goodies (soft of course) inside with the bike and be mindful of the airlines weight limit lest you get held up at the airport trying to unpack that thing. Have a Camelback or something similar. The heat was brutal and water needs were excessively high. On the climb, many onlookers will give a passing cyclist water and cheer you on, just as though you were in the Tour (which was fun), but I had some 50 miles or so to go to the base of the mountain, and the ride back to do as well. Sunblock is a must. It was brutal up there that day.
In a car, I havent too much advice I am afraid. On a bike you can ride around up there until they motion you off the road, and that was only moments before the lead rider came through. A lot more sitting and waiting in a car. Either way, its a truly memorable day.