you can get to North Carolina from Chicago by train, but it is a long and convoluted route.
Starting at Union Station in Chicago, you'd have to go to Washington, DC, wait 17 hours, and then go to Greensboro (you didn't specify a city in North Carolina, so I'm just assuming this is where you're going).
You could also go from Chicago to Charlettesville, VA, but the wait in Charlettesville is even longer - 20 hours.
The train (the 30 - Capitol Limited) has several stops from CHI to WAS, so it's also a 17 hour train ride, a 20 hour wait, then another 6 hour train ride from WAS to GRO.
ticket cost for no advance purchase leaving today is $161.
And, you can also go by bus - from Chicago to Greensboro, NC, there's a bus that leaves at 2:15pm, takes 1 day + for the trip, then arrives in Greensboro the next day at 5:10 pm. - $126.
If you can fly - American Airlines and United Airlines have non-stop flights from Chicago to Greensboro. No advance purchase, one way, is $220. And it's only a 2 hour flight.