Hi Ally!
I'm writing from my campsite in Les Bossons, about a mile down the road from Chamonix. I have been pushed off Mt Blanc three times in the last ten days by bad conditions. The very first thing I should say is that if you want to climb the mountain, especially unguided, then aim to come here for at least two weeks, if you can book a month, even better! The single biggest reason for not succeeding is the weather. For example, three times now I've been pushed off by extreme winds, lightning, avalanche and rockfall risk, and visibility down to 5m making navigation impossible. This has been a very bad year for Mt blanc so far, the path from Nid d'Aigle to the Tete Rousse glacier...there is no path! There is almost a complete blanket of snow right down to 2300m, so right now, you either need a guide (and no cloud or wind), or you need to be n excellent navigator, with little interest in self preservation.
At the same time, it's perfectly possible than in two weeks, all that snow will be gone and you can see the path. All you need then is a good clear day and anyone with no experience in track shoes and jeans could hike all the way up to the Gouter.
So the technical difficulty of the climb is very low, but what I see here all the time is people who are only here for 1 week or even a long weekend trying to force a way up a very difficult and dangerous mountain route, as are the current conditions. So I would say with your experience to make sure you come here for long enough to wait for good weather and snow conditions. If you get to climb in the first few days, then no harm!
There is absolutely no need, as suggested, to fork out hundreds of euro for so called 'mountaineering' courses. If you do a LOT of studying on your own (there's hundreds of books from which you can learn all the technical skills you need, as well as studying mountain weather, avalanche awareness, crevasse rescue, nutrition etc), as well as a lot, and I mean a LOT, of hiking with a heavy pack and as much height gain as possible, then all you need to do is come to Chamonix for a month, find partners (easy, people everywhere happy to help) and go out for a few sunny days on the Mer de Glace to practice everything you've learned. Then you'll be ready! If however, you are lazy and don't want to study alone or find partners to practice the essential skills with in a safe place, then by all means spend all your cash on courses

Or you can do what lots of others do, take the fist tram or cable car in the morning and follow behind the guided groups. Not good when you lose them though.
I'm lucky that I'm here for the season, so while others might succeed in forcing a way up, and happy in the knowledge that I can return to the mountain when conditions improve. In the meantime, I can hang around in the campsite slackining etc, or go rock climbing, ice bouldering. I'm telling you it's the way to go.
Good luck!