I'd suggest looking at the tracks nearby where you may be doing this and see what organizations put together open track days. In many cases, this will be done by NASA or SCCA. There may be some other freelance rentals or clubs that allow it, but the majority will probably be by a sanctioning body of some sort.

Look through these organizations sites to see what options they show for these track days. For NASA, they offer the HPDE events for simply bringing your car out and driving it on the track. These are primarily geared towards using street cars, not dedicated track cars, although those do show up on occasion. The next step up from that would be their Time Trials events. This is actual racing against the clock with several different brackets of cars and their mods that they use to determine classification. SCCA has their own set of classifications, as well as a set of new rules they have developed toget the pro-touring crowd involved with a slightly less regulated class called Classic American Muscle.

Since you may be years out before beginning this, find the organization, find out who the local contacts are, visit some events to see what they look like, and talk with some of the officials about how they deal with a track specific *race* car, decide if you want to formally join their group. This will give you the background to decide how you want to proceed with a build up of a car as well as possibly what car you want to pursue.