Cool
Hi Erwin,
I think you can take your rental there.
During "open driving" (Touristenfahrten) times, it is not a race track, but public road. Even tour buses supposedly go around there at times.
BUT, you can only go at certain times, they are listed here:
http://www.nuerburgring.de/en/nuerburgring-race-track/hours-of-operation-nordschleife.html?no_cache=1
Some tips:
If you don't know your way around the 'ring area, try to research beforehand where the entrance to the Nordschleife is, since it is not well labeled.
There may be some traffic congestion at the traffic circle at the entrance, don't worry, you will get there and the Nordschleife will not be as crowded as you may think.
I recommend you get 2 or 3 laps. There is a machine (like a parking ticket machine) where you can get tickets. Decide how many laps you want and get the tickets for that amount of laps. Don't plan on parking and buying another round - big hassle.
Two laps is good and may be enough, don't get more than three.
Use the first lap to get a feel for things and the second to go fast.
Stay right in areas where you are slow or to let others pass, but don't let traffic make you nervous. Do your own thing.
HAVE FUN!