About all you need are:
* money
* a car or similar vehicle
* more money
Once you are in the U.S., visas are not needed except near the Mexican border (30 to 50 miles) where there are immigration spot checks, especially in Texas and Arizona.
Do not underestimate the size of the U.S. Driving flat out/non-stop from NYC, it is 75 hours to the west coast (Seattle), if not stopping for food or fuel. Doing it in a week is quite rushed and rather miserable to endure. Most people take a few days to stop in interesting places, especially the fantastic national parks in the Rockies: Yellowstone, Grand Canyon, Zion, Glacier, Teton, Bryce. The first three are only superficially experienced in a one day visit. To do justice, plan on at least two days at each. A comfortable once-across-the-country road trip with lots of visits to interesting sites would be in the 10 to 14 day range, though you probably wouldn't get tired even after 30 or 40 days. (More likely to run out of money.)
If you have money to burn, you can rent ("hire") an auto. As another answerer suggests, you'll have trouble arranging that with the major rental companies if you are under 25. This is less likely to be an obstacle for a UK national (U.S. drivers 16 to 25 are the most threatening demographic for an auto). There are companies like http://www.rentawreck.com/ which are far more friendly to 18 to 24-year-olds.
But if you'll be staying in the U.S. before or after, it probably makes more economic sense to buy a vehicle. You don't have to be a citizen, though auto insurance is likely to be a significant expensive. Check your local classified advertisements for a well used vehicle, and shop around extensively for insurance—there is a huge range of rates even for basic coverage. If you are in the west, there are unlikely to be problems with rust (caused by road salt) like there are in the north and east; a good used car can be as little as $2000 to $4000, but expect to put in another $500+ to make it cross-country worthy. In salt-road areas, inspect carefully for structural problems. After driving it for a year or two, you can likely sell it for almost as much as you paid for it.