Chamonix is such a popular destination that getting there and back is very straightforward, most people fly via Geneva Airport and arrange one of the many taxi/minibus transfers to Chamonix or its satellite towns and villages. The complexity and planning starts to have an impact if you don't want to be hanging around in Chamonix for an extra day while you get supplies etc, which we didn't. At the very least you will need to buy fuel for your stove before you set out on your trek and this means having some time free when the shops are open.
I left home at 6.30am and was dropped off at Newcastle Airport at 07.15 ready for my 9.00am flight. I used a Green Hermit Waterproof DryBag rucksack as hand luggage in which I carried my Sleeping Bag and Sleeping Mat, Tent, and electrical gear and my camera. I usually do this as it ensures that if my hold luggage goes missing I have with me the key items that would be difficult and expensive to replace. I also travel in my hiking gear, including footwear as I don't see the point in having separate travel clothes when you are backpacking.
My BA flight left Newcastle on time and arrived at Heathrow at 10.00am. My flight to Geneva left at 12 Noon and arrived at 14.15pm. I had pre-booked our transfer and we arrived in Chamonix 90 minutes later, in good time to go shopping. We'd booked a hotel for our first night, which proved to be very useful as we could drop our bags off then go into town. It also gave us a good start in the morning, which was very wet.
Getting back was a simple matter of reversing the outward journey. We got picked up outside our Chamonix hotel by the transfer company and the rest of the journey dropped into place.