Helsinki to Cologne

It will be quite difficult to travel only by rail (since you'll have to go all the way via Russia), but if you are ok with traveling some parts of the route by other means of transportation, then it can be accomplished. Here are few option for you to consider:
a) ferry Helsinki-Stockholm, then high-speed train to Copenhagen, from there train to Hamburg and finally to Cologne (or alternatively overnight direct train Copenhagen-Cologne). More or less comfortable and fast way to travel, but won't be really cheap. In regards to travel time: ferry Helsinki-Stockholm takes roughly 15+ hours, then train to Copenhagen is another 5 hours and finally train from Copenhagen to Cologne is 12 hours (direct overnight) or 9 hours (daytime with change in Hamburg). Regarding fares it is difficult to say without knowing how far in advance you can purchase ticket or whether you willing to use discounted non-refundable/non-exchangeable tickets.
Ferry companies are Tallink and Viking Line, fares for Stock.-Copenhagen can be checked via Swedish Railways website (sj.se), rest via German Railways website (bahn.de)
b) ferry Helsinki-Tallinn, then bus to Kaunas, then train to Warsaw and finally direct train to Cologne from there (also possible to win some time if you travel all the way to Warsaw by bus). Because of possibility to purchase discounted ticket from Warsaw to Cologne this option can be rather cheap (ferries to Tallinn and buses in Baltics aren't very expensive), but it is quite time consuming. Travel time is 2-3 hours for Helsinki-Tallinn ferry, then roughly 9-10 hours to Lithuania, another 9 hours to Warsaw and finally 11+ hours by direct overnight train to Cologne (or 10+ hours during daytime with change in Berlin). Again with train portion of the journey ticket fare will depend on how in advance you can buy your tickets.
Bus companies in Baltic states are LuxExpress, Ecolines, SimpleExpress, and again DB website for Warsaw-Cologne leg
c) rather cheap might be an option simply to fly most of the distance, especially using low-cost airlines. But here it's good to know either exact date of your travels or at least how far in advance you plan to travel, since most of the cheap tickets need to be purchased ahead of time. Just as an example, Tampere-Bremen with RyanAir (Helsinki-Tampere and Bremen-Cologne done by train), or Laaperanta or Tallinn to Dusseldorf Weeze etc.