Getting to Chamonix from Geneva airport will be a lot easier very soon.
Tourists will be able to use a new train line, which will connect Geneva to Saint Gervais train station Le Fayet. Then, The Mont Blanc Express will get you to Chamonix.
This project should be completed until 15th December 2019.
From Geneva airport, you will have to take a train to Cornavin (RegioExpress). From there, you take the Léman Express to Saint Gervais Le Fayet.