By car:
http://www.pressoffice.uni-koeln.de/anreise_auto0.html

By plane:
The airports to fly into are: Cologne, Duesseldorf or Frankfurt. If you fly via Frankfurt, it might be easier to take the train from Frankfurt Airport to Cologne, rather than flying from Frankfurt to Cologne or to Duesseldorf. The train takes 50 minutes and takes you right into Cologne Central Station. Some airlines allow booking the train as a 'flight.'
http://www.pressoffice.uni-koeln.de/anreise_flug0.html