If you are going with somebody else, you can just manage it depending on where you fly to. If you are going alone, forget it - with the exception of the flights and food most of your costs are halved on a per person basis when there are two of you. It's no cheaper to book a hotel room or hire a car for one person than it is for two.
As a guide...
Half decent car is £150-£200 a week. You don't want to road-trip in an Economy Rental, US road trips in Yaris Saloons are totally crap. £150ish a week should just get you into a regular fullsize Sedan, Dodge Charger or similar.
Fuel will cost you on average £100 per 1000 miles driven (You might do it for less with a more efficient car, more with a more thirsty car but this is a decent average based on about 32mpg UK).
Accomodation you can average 50ish quid a night depending where you decide to stay.
Food can be cheap if you like takeaway junk - a large dominos is usually under $10 for example.
Flights - good value would be £400 to the East Coast, £500 to the West Coast. Lots more expensive if you go in summer. Pick a route with loads of competition for best value flights - cheapest East Coast destination is almost always New York, West Coast is LA. If you pick a city that's not on a popular route you will pay dearly for it which is hugely annoying, my 9 hour flight to the Pacific Northwest cost me £300 in economy more than I paid to get to LA - 11 hours - at exactly the same time of year 3 years previously. Pay more for less, amazing, grr.
I recommend June or September as the best months. Both are shoulder season months so cheaper flights, much cheaper than July and August, but you still get the weather. Avoid June if your route involves mountain passes as they may be closed.
Just on the train home from my 3 week road trip now. Sad times
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