I haven't been to Silverstone before but i have done a lot of research on the options and spoken to some 'veterans' before making up my mind. That said, some of the others here will know something i don't on easier/cheaper ways to get to the track.
My tuppence for what its worth:
If you are absolutely set on staying in London on all three days i would recommend a hire car and buying a silverstone car parking pass (got to have more than 1 person in the car).
Public transport will tie you down to certain train times, meaning you can miss some good parts of the day, can be expensive and doesn't run at convenient times. I am advised that on Grand Prix weekends taxis from the station to the circuit can range from £15-£45 (not sure if this is true).
The nearest train stations are northamption and banbury. you can check train times on
www.thetrainline.com or coaches on
www.nationalexpress.com. The first train on a sunday leaves Marylebone at 8.40 and gets into banbury at 10.50. An open return ticket (so you can get on any train home) will cost around £60. You can get fixed time tickets for about £25. So with tickets and taxis you're perhaps looking between £40-£100 per day travelling expenses
If you decide to go down the hire car route remember London is 70 miles away from silverstone, so a fair amount of travel will be involved everyday and petrol is currently ranging between 90.9-99.9 per litre in the london/banbury areas.
Alternatively
http://www.motorracinginternational.uk.com do coach trips to the circuit for race day and you don't have to buy your tickets through them. You pick up the coach at a pre determind point and they take you ino the circuit and home again. This is only for sundays though.
Silverstone does run a free park and ride bus from certain junctions of the M40/M1 where you park up at one of the services and they bus you in to the circuit. Sadly this is only for the sunday i think.
To be really honest i think it would be far cheaper for you to camp in or around the circuit for three days. Campsites vary between £60-100 for all three nights. Some people have said that some of the food vans in the camps are relatively inexpensive and there is a supermarket reasonably close by for you to stock up on nibbly foods and drinks.