Train will be your best bet, TGV via Lyon then possibly Geneva.? Taxi/transfer link up for last bit. Go on the sncf website. Bike hire possible in Chamonix 🙂
Edit2: quick search – Nice Geneva or Annecy 7hrs and £70pp (family tockets available too I would guess), Nice Lyon 4.5hrs £55pp
As an aside its further to drive than you think, if you look at a map Chamonix is as far from Nice as is Paris (so a 7hr drive). We’ve driven Paris to South coast a few times and we generally stop overnight.
Edit: just seen its for a family .. hum, more tricky as cost of multiple tickets quickly adds up. A car in Chamonix can be useful, depends on what you’ve got planned. The train can get you around the valley and is free with accomodation, not fast and shuts for 2hrs at “lunchtime” mind. If you are staying centrally you can walk to most of the things you want to do. when bikingbthere we used the car a lot as some routes / places are much easier / faster to get to (eg Les Hoiches)