It is quite easy to travel from Düsseldorf Airport to Cologne by train.

Düsseldorf International has two train stations. There is a commuter train station (S-Bahn) right at the terminal. The long distance trains depart from the long distance station, which can be reached from the terminal by the so called skytrain (take the escalators up) or by shuttle buses if the skytrain is not working 8due to maintenance).

There is a ticket office of Deutsche Bahn. You could look after trains at the homepage of Deutsche Bahn by clicking on internat. Guest right at the top

Note, this is not Düsseldorf Main Station. If you like to travel to Düsseldorf Main Station before heading to Cologne, take the Commuter train to Main Station, ticket boxes are at the entrance to the platform. Between Düsseldorf and Cologne is a frequent schedule of trains (from regional train with several stops to High Speed by ICE).Anyway, travel time is app. 20 minutes by train (ICE, IC, EC) or some 30 minutes by regional train.