My girlfriend and me want to go from Cayenne (French Guiana) to Europe by boat/ship/vessel. Is there anyone who has some information about it? Where can we find a boat? Do we have to go to the harbour and ask around or are there any web pages to look up before? How much would it cost?
In addition to Cargoship Cruises provided by @andra there is another website that provides similar information Freighter Cruises operated by Maris which upon close examination has a Cruise on a CGM CMA Freighter between Europe and South America(see Blue Route). If you look at the list of stops one of them is in Cayenne, French Guiana, which is what you're looking for.
One thing to mention. If money is no object you can try to rent a Yacht and cross Atlantic in Style.
Ok, renting a yacht is too expensive. Even to go by cargo boat costs a lot of money. In French Guyana operate only two shipping companies (CMA-CGM and Marfret). Marfret does not carry passengers. CMA-CGM carries passengers, but its a cost of 100€ per person and per day. A vessel needs 23 days from French Guyana to Rotterdam/Netherlands (via Brazil and Spain). It's a total cost of 2.300€ per person.
An other option to cross the Atlantic Ocean is by private sailing boats. But because of the sea (wind and currents) the season is from April to June. In the other months the sailing boats leave to Europe from the Caribbean Islands like Martinique. So you have to go there first.