Of couse you can go by car. 1 hour from Geneva. Drive to Chamonix. 

Hi Mehar Shah,

If you mean the mountain, then what Nir says is of course spot on. You can also take the Cham Express bus from the airport http://www.chamexpress.com/ (all times and costs are on the site in English). It stops in Chamonix centre or your hotel.

If you mean using the tunnel to cross into Italy - il traforo del Monte Bianco then you will need a car or motorbike for sure. Here is the site : http://www.tunnelmb.net/v3.0/  it costs EUR 43.50 one way or EUR 54.30 return in a car. Slightly cheaper with a motor bike.

Hope it helps,

BCRover

Train is possible from Annemasse but takes a long time.  There are also buses from the gare routière which might be cheaper than from the airport.   If you do not mean Chamonix then it depends where you are going as different resorts villages are in the area.  If you are meaning the top of the mountain, it might be still possible to go up on skis otherwise I would assume the alpinist season starts around end June (but from the wording of your message I do not think you are intending to go up the mountain