Currently, I have created a game, which shares data with a server backend. This server also validates and stores player data, for example high score.
Is it possible to send scores to GameCenter via my server backend? If yes, how?
Sure, I could send the high score from my server to the client and then let the client push the data to GameCenter, but this would make it easier to cheat.
In short: no.
Access to Game Center has to go through the GameKit.framework which is only available on iOS and OS X.
Furthermore, players need to be authenticated. Therefore a single server can't send scores etc. on behalf of the players even if it were running on OS X because only the server's identity could be used to send and update scores etc. So even if your server were OS X and you had a proxy app communicating with the client and Game Center it wouldn't work because there would only ever be one client sending scores. A leaderboard with one entry (the best score among all players) wouldn't be of much help.