National Express coaches are probably the cheaper and slightly easier option (approx 3-4 hours, traffic-dependent; they go round London on the M25 motorway), if not quite as fast as train plus subway through central London (approx 2.5 hours; below). This website has coach times and prices:
http://www.nationalexpress.com/home/hp.cfm
The coaches to Cambridge seem to run every 1-2 hours. The coaches terminate in Drummer Street, Cambridge, which is central, and about 10-15 walk from Clare College (if you know where you're going), or 5 mins by taxi.
The alternative train and subway option through central London is not difficult either: Subway (aka the "Underground" or "the tube") direct from Heathrow to Kings Cross rail station (on the "Picadilly Line", about 50mins), then overland train direct from Kings Cross to Cambridge (about 60mins). This will be more expensive, and slightly more hassle if you have lots of luggage (ie more dragging). Tubes run every few minutes; trains are also very frequent, with timetables here: http://www.nationalrail.co.uk/
[Yet another alternative is the "Heathrow Express" overland train, but this goes from Heathrow to Paddington station in London, which isn't as handy for Cambridge (ie you'd still have to get the Tube from Paddington to Kings Cross)]
Finally, these websites may also help: