Most people use accommodation to complete the Tour of Mont Blanc, if you want to backpack the route there aren't camping options 'at every turn', a bit of knowledge is useful here so I've summarised the camping options I encountered.
A common starting point is Les Houches, there's a basic campsite here and you get access to supermarkets for provisions not far outside the environs of the campsite. The Geneva airport transfer service will also drop you off here which keeps things very simple, except for getting fuel.
The main shopping area of Les Houches is about twenty minutes walk and I can't confirm if gas is available because I had no reason to investigate. Most supermarkets sell camping gaz but you need a mountain shop to get the screw in gas cartridges for most stoves sold in the UK. The Frech equivalent of Meths is usually available in supermarkets also, ask for 'Alcool a Brulee'.
If you need screw in gas cartridges and there are none in Les Houches it means a trip to Chamonix - where they are definitely available. If you are worried about getting the gas you could get dropped off in Chamonix by the transfer company before getting a bus to Les Houches - the buses are frequent and cheap.
If you take the high variant from Les Houches then the campsite to the South of Les Contamines-Montjoie is a good option, if a little family orientated.
If you take the standard route you will have more time - there's an official wold camping spot near to Refuge Nant Borrant.
If you're really fast you can hike up to the Lacs Jovet where wild camping is available.
There isn't much opportunity to camp on the route until the descent from Col du Fours, assuming you are doing this variant (having done both I would highly recommend this as the normal route is just that - normal). If you descend all the way to La Ville Du Glaciers you will end up camping at the Car Park - which I didn't like - much better to wild camp on the descent - don't get too low - if you look around and see flat areas to camp stop where you are - if you go too low there aren't any suitable sites.
If you thought you might camp outside Refuge de Mottets don't - its not allowed. Although there is a possibility above the Refuge if you carry water up with you, there's a small shelf where you could camp out of view of the Refuge but you can be seen easily from the path.
There are places to wild camp near to Refuge Elisabetta but this isn't far from Refuge Mottets and you will probably be heading for Cormayeur. If you want to stay on the route there are wild camping spots near Refuge Maison Vieille if you stay out of sight of the Refuge.
If you want a hot shower then drop down to Val Veni and try the campsite Monte Bianco which comes highly recommended. There are no campsites in Cormayeur.
The next day is limited for camping although there are some wild camping spots directly above refuge Bonatti - again, so long as you are out of sight camping is tolerated.
The route takes you down to the valley floor then up over Grand Col Ferret. You could wild camp on the Swiss side of the col at a push but better to push on. The route goes through a farm, not sure if they allow camping as such but you could stay in one of the Yurts that they have available - which is almost camping - just not as good!
If you can push on then camping is prohibited until you reach La Fouly. We found a good place long before this, just before Ferret, but there are lots of signs prohibiting camping. There are no campsites in Ferret.
From La Fouly you have two choices, the campsite at Champex or about an hour further on at Relais d'Arpette - both are good.
Whether you cross the Fenetre d'Arpette or take the low level route there's the basic campsite at Le Peuty.
There is a wild camping option on Col des Possetes though I'm not sure water will be available all year round . The obvious place to stop is Trelechamps, there is accommodation but I don't think there's any camping.
Camping is allowed outside of the Refuge at La Flechere, ask permission but it will be granted for the lakeside.
You should be able to return to Les Houches. However it would be a shame not to visit Chamonix while you are in the vicinity and I would recommend the campsite at Le Praz de Chamonix.