Your last question seems to indicate you think there's some kind of short-term statute of limitations or something or that the law would require they notify/bill you when you bought the car. I wouldn't think that's true (most people know whether their county charges a car/personal property tax, and the DMV and the county are two separate entities).
Can't know from here what the tax period is, but if you moved out of the jurisdiction, you can expect to report that and pay a pro rata/proportional share of the tax (your current county may have the same thing). I would like to think you read up about this before posting here, but if not:
http://www.cookcountygov.com/portal/server.pt/community/revenue,_department_of/313/use_tax/487