As far as it seems you have two major option here:
1) Airplane+train: Ryanair operates Bratislava-Frankfurt (Hahn) flights three times per week. Frankfurt (Hahn) is rather far from actual city of Frankfurt, but nevertheless you can manage it rather easy with special bus which will take you from airport to the city in 1:45-2 hours and 12 euro (
http://omnibusse.bohr.de/routes/wtc/routes.php?action=detail&language=de&route_id=16). Kosice-Bratislava route has quite good rail connection, it takes from 5 to 6 hours and from 17 to 18.5 euro (depending on type of train).
2) Train: Here you have as well a number of choices
a) via Bratislava, Wien: Once again take a train to Bratislava, from there train to Wien run really often and cost rather cheap, but unfortunately most of them arrive at Wien-Sud., while ICE trains to Frankfurt depart from Wien-West. So you will have to change in Wien or use public city transport. From Wien you can get cheap discounted Spar ticket to Frankfurt for as low as 29 euro if you purchase it advance here -
https://ticket.oebb.at/bin/frame_ticket.pl?ticket=jticket.pl?ABBFSEL=8100003&ANBFSEL=8000105&LANG=EN b) It might sound strange, but fastest and app. the same price option is to go to Frankfurt via Budapest. It will require you to make only two changes in Budapest and somewhere either in Wien or Linz. Kosice-Budapest train 6:13- 9:13, Budapest-Munich train departs 11:10, arrives at Wien West at 14:08, while Wien-Frankfurt train departs Wien West at 14:40. DB website suggests to change in Austrian Linz and not Wien - arriving to Linz at 15:50 and departing 16:16. And here once again you can purchase Spar ticket for the whole route from Budapest to Frankfurt for 39 euro (online and in advance from DB website).
Most reliable website for all the timetables -
http://reiseauskunft.bahn.de/bin/query.exe/en?newrequest=yes&protocol=http:& In case you have any additional questions - feel free to ask