How and where to get Chinese visa if I live in Lincoln Nebraska?

You should apply China visa through Embassy of China in Washington DC because Lincoln Nebraska is under its consular jurisdiction.

You may submit the application to the Visa Office of the Embassy If you cannot come in person, you may entrust someone else or a travel/visa agent to drop off your application at the visa office of the Embassy. No appointment is required. Mailed applications are not acceptable and will be returned.