I don’t know why you are talking about phpbb and bbpress?
The actual problem does not involve forum software, does it?
Something that I found is following:
http://codex.wordpress.org/Author_Templates
This explains how to create your player profiles. Create a author.php in your theme folder and make it display any author related information you want. You can also run mysql queries and format the results properly to display your gaming stats.
As Ahmad mentioned, it is reasonable to store the user_id of your gamestats mysql table in wordpress user meta data.