Flying

It's a very short flight (1 hr 15min - 1 hr 25min) from Britain to Geneva. Plenty of airlines fly on this route, many of them from the small and efficient London City Airport.
From Geneva airport, it's just 100km to Chamonix via rental car, airport transfer or bus.

Airlines

http://www.swissair.co.uk - often has good fares and has regular flights from London City Airport
http://www.airfrance.co.uk - has just started flying from London City Airport and has good prices.
http://www.ba.co.uk - the Gatwick routes are often very cheap
http://www.easyjet.co.uk - Fly from plenty of regional airports
http://www.jet2.co.uk - Low cost flights from Leeds, now all year round

Flight Agents

http://www.skyscanner.net - excellent fare-finding engine, particularly for Easyjet, allowing you to see the price fluctuations.
http://www.traveljungle.co.uk - Good all-round fares
http://www.ski-flights.com/genevaflights.html - Ski flight specialists

Geneva Airport

Make sure you check which side your flight arrives and departs from - most are from the Swiss side (Secteur Suisse) but occasionally you do get them (notably with Easyjet) from the French side (Secteur Francaise). There is a short walk between the two sides which is not as well signposted as it might be, so it's best to check - and if you're hiring a car, the same applies - see Car Hire for more on this.

Geneva airport is generally very efficient, as you'd expect. It has good facilities including an excellent nursery, extensive duty-free shopping and a rather odd "Lost" style plane fuselage upstairs which children can sit in.

Once at Geneva, you are just 100 km from the chalet, via (in approximate order of convenience and expense) hire car, airport transfer, bus, or train.