I went to school five minutes from Silverstone.
If you insist on training it, then you need to go from Euston to Milton Keynes. Then it is about a half hour drive to Silverstone by cab (plentiful) from there. MK is the best station as the trains are InterCity (ie fast, regular, not too many stops en route and more comfy than the local trains which you are more likely to find on the Northampton run).
I would not recommend this public transport option as you would be paying for Heathrow Express, tube from Paddington to Euston, rail from Euston to MK (likely to be expensive if you don't purchase online ahead of time, whcih means fixing your travel times, whcih is tricky), plus cab from MK to Silverstone. On top of all the the hassle of lugging your stuff along, dealing with the heaving unwashed masses, it would take three to four hours all in, and you would end up with no transport when you get to your hotel (it's a nice part of the world to meander on a country drive).
Being from NYC, you may not be keen/able to drive yourself (which may be the reason for the need to train it), but I would stongly second the hire car from LHR option. Just request an automatic, and ask for Sat Nav. It should take about an and a half to drive there (it would take me an hour as I know the route), just zoom out on the M4 then M40, come off at the Bicester exit and follow the brown signs to Silverstone.
If you are uncomfortable driving, the other option is to hire a driver or even get an airport car service which should charge about a hundred dollars for such a trip, one way.
Do visit my almer mater
www.stowe.co.uk - one of the finest Palaces in the world, once owned by the Dukes of Buckingham while you are there.
Your conference will probably have organised somewhere for you to stay; if not I can recoomend