You can try european timetable at: http://reiseauskunft.bahn.de/bin/query.exe/en?ld=104&newrequest=yes): The route is Copenhagen - Helsinki -St.Petersburg by train. Unfortunately I have absolutely no idea about costs but can advise about the duration of such a journey: 20+ hours to Helsinki, than 6,5 hours to St.Petersburg.
Try another useful link http://www.seat61.com/Finland.htm#Helsinki%20-%20St%20Petersburg%20by%20train, it's devoted to travel from the UK through DEnmark to Moscow and St.Pete's - tons of information with all necessary contacts.
As another option take bus ride from Copenhagen to Stockholm (385 DKK), 9'45 one way, than overnight ferry to Helsinki (departure 17.00- arrival 9.55), rate starts from 855 DKK one way and the last stop to St.Pete - by train (departure at 17.00, arrives 22.51) paid 55 euro the cheapest, by regular bus - almost the same (38 EUR), starts at 11 p.m. meaning that you'll spend one more night onboard. The cheapest option to get from Helsinki to ST.Petersburg - take tourist bus at about the noon, rate is 20 Eur or even less depending on availability. Detailed description is here http://www.skazka.no/travel-to-spb.html (written 5 years ago but yet actual)