What the cheapest mode of transportation is at a given time largely depends on the time of booking and the available tickets at that time. There are many possibilities to get from Paris to Dusseldorf.
Most of them are already mentioned:
- You can take the Bus (Eurolines)
- You can use a major airline (use a meta-search engine such as Kayak to search for many possible combinations of airline, departure and arrival airports ("proximity search")
- Low-cost airlines that are not always listed in the meta-search engine output
As a rule of thumb, don't focus on the low-cost airlines. Sometimes they are more expensive than the major airlines. Also, Wikipedia is an excellent source to search for possible combinations of airlines and arrival airports. For example, it is written that Germanwings flies from CDG Airport in Paris to Dusseldorf: http://en.wikipedia.org/wiki/Charles_de_Gaulle_Airport - don't forget to check the other Paris airports as well!
And don't forget to check the trains as well. As written in the comments already, the trip is actually rather short. Check the websites for the Thalys, of SNCF (French main railway operator), and DB (German main airline operator). They all offer different prices and different connections. One may suit your budget. Especially if you consider that you also need transportation to and from the airport, the train may be an economic option.