In reply to HBBS:
Bivvying is tolerated by Albert Premier, sometimes see tents there too. Loads of good sites above and beyond hut. Tap outside hut for water, loos accessible.
Some years you can go all the way up the Mer de Glace into the Vallee Blanche to reach the base of the Midi - completely conditions dependent. Early in the season there are some routes up the Midi from the Cham side which may be in condition. Check the guidebooks. There is always the Frendo Spur but I personally would see that as a separate route, not the first day for a Mt Blanc ascent.
As others have suggested acclimatise perhaps from Albert Premier base or around the Domes de Miages. Then do a 4,000m peak such as the Tacul ( lots of route choices ) using the lift and an overnight hut / bivvy or two to acclimatise further. Or you could head to Gran Paradiso or Zermatt for some further easy acclimatisation and different areas. Check conditions before you set off. For example, the normal route up the Tacul can be a pleasant stroll (albeit with seracs), an avalanche trap or even need some steep climbing skills to get past crevasses / bergshrunds.
Some people walk from the valley to the Tete Rousse hut, then another day to the Gouter followed by a summit day. Personally, I prefer the Trois Monts route and a descent via the same way or via the Gouter.