I've never made this journey myself so maybe someone else can advise better, however renting a car at Heathrow and driving could be the best option;

Travel Planner at www.theaa.com advises 70 miles to the circuit from Heathrow, taking 1.5 hours

http://www.transportdirect.info/ which is a government run website that can compare various modes of transport advises that the only option to the racing circuit is to drive ! However if you are visiting the town of Silverstone itself then there is an option to travel by train from London Euston to Northampton and then by bus to Silverstone - which would take 2-3 hours in total..

Another useful source of info is at http://212.188.131.251/scripts/webtr...r.dll/journeys

If you can tell us if you are travelling to the circuit or the town and whether it is during the Grand Prix then perhaps someone else may be able to help more ?