- find a friend of a friend who needs a car moved. This is ideal.
- post on a rideshare board, like erideshare.com, saying that you're looking for a driveaway.
- post on craigslist or something; I never got this to work so I don't know how's best to do so.
- go through a company called Auto Driveaway. They maintain (to put it generously) this site, and if you find a car you want to drive, call them up and work out the details.
The friend of a friend is best, because if you work this out with a stranger, there's no real official way to do it, so you kind of have to trust each other. The car owner's insurance will cover the driver, and you could try signing some kind of contract to work out the details, but no guarantees.
Auto Driveaway is great in theory, but in practice it's a bit sketchy. I noticed 3 cars listed online, so I called them up to find that those 3 were gone but there were 2 more available, and I agreed to drive one to Chicago. A few days later, I called back to find that that car was unavailable (a week before I planned to leave), but I could try to call back on Monday about another car going to Virginia.
More details about working with Auto Driveaway: generally, the driver has to pay the gas. They'll let you drive any route within 20% of the most efficient route (so if you're going Seattle-Chicago, which would be about 2000 miles direct, you can go 2400 miles) and they'll let you take up to 10 days. And if you're calling up Auto Driveaway Seattle, Tiffany is super friendly; Scott is rather... direct.
If you're willing to deal with the sketchiness, unafraid of having to deal with them if the car breaks down or something else unfortunate happens, and super flexible with your travel dates (i.e. you can just pick up and leave whenever), maybe doing a driveaway would be great! As it is, I am not.
This is a long way of saying, I was going to do this epic road trip, but now I'm instead going to do an epic train trip. Thursday-Saturday on the Empire Builder to Chicago, stay a couple days with Dave and Erik in Chicago, and then on to Cleveland on Tuesday. The wheels are getting in motion! The ignition sequence has begun! All around the world, here we go!