My husband drove the Nurburgring Nordschliefe in 2009. It's the 20.8 km, 12.9 mile Nordschliefe you can actually drive. This is a link to it: http://www.nuerburgring.de/en/drives-fun/drives/touristenfahrten/tourist-rides-nordschleife.html. The link should give you lots of info on going.
When were there, I saw everything from vans with firemen in them to motorcycles to vintage Mercedes convertibles. This is a little video that shows that: https://www.youtube.com/watch?v=43FZqZEUKK4.
This video guides you through the track from start to almost finish, and may go faster than you will, but it does show the corners: https://www.youtube.com/watch?v=uWW0epQtOfg. Imagine doing this with all the other vehicles you saw in the previously linked video, not in the orderly line shown in this one. You can find many more of these by doing a Google videos search for "driving nordschliefe."
This is a map of the track with the danger points: http://www.nuerburgring.de/fileadmin/Streckenskizzen/NordschleifeGefahrenpunkteEN_2013.pdf.
There are restrictions on rental cars and we were told that rental agencies patrol the parking lots to make sure none of their cars are there. My husband rented a race-prepared Suzuki Swift from Rent Race Car: http://www.nuerburgring.de/en/drives-fun/drives/car-rentals.html. They have the costs down to a science, which you can see at the link. We left our regular rental car at their office. Things have changed a bit since then, I'm sure since I see on the website that you can pick up the race car right at the entry to the track.
Whatever vehicle you use, be aware that there will be both slower and faster vehicles than yours on the Green Hell. It requires some skill.
Also be aware that you can't just go anytime you want, and that even if you plan to go at an open time, the schedule may change. This is a link to current scheduling here -- http://www.nuerburgring.de/en/drives-fun/drives/touristenfahrten/tourist-rides-nordschleife/opening-times-nordschleife.html -- where the fine print says "Please note: Daily opening hours vary! The exact opening hours can be seen by moving the cursor over the calendar days (or further below in the table of opening times)."
Bottom line, you really need to explore the website to learn when you can go, how you can go and how much it will cost. At first my husband thought he would be able to do only 4 laps and that's what he paid for, but he later decided to continue driving and did a total of 8. If he'd decided to do that many from the beginning, it wouldn't have cost quite so much per extra lap.
Full disclosure: my husband hobby races a Formula Ford and he did prepare for the driving before we went on that trip by doing behind the wheel practice sessions sitting at his computer at home. Driving the Nordschleife was a bucket list thing for him, but I bet he'd like to refill that bucket again!