The game needs to have a leaderboard that shows scores for everyone who plays and wants to record their score for this game, so that the users can be motivated to play to get a higher score. This is where game center comes in. The user should be able to connect to game center using a login for game center, and/or facebook. Also, have the option to put in an alias for a single gaming session.
When the user clicks on the trophy in the game, the top users scores are shown. When the user scores their own score, then the leaderboard will focus in on their score and the scores nearby them.
Their score should be highlighted, indicating it's their score.
There should be a popup asking if they would like to record their score on the leaderboard, after they stop their gameplay.
If they say "yes", they will have an open field that asks them to type in their name or alias. if they are logged into game center already, the name should be populated automatically.
An "ok" button or a "cancel" button should be available.
After they hit "ok", then the leaderboard will show where they ranked relative to their peers.
They should be able to scroll up and down to review other users names and scores.
After they're done playing and they see where they are on the leaderboard, there are buttons to play again, to exit, or to go back to home.
I've provided screenshots with some directions and some sketches as storyboards. They should be intuitive.