As you progress through the career mode of the snooker mode you will be invited to tournaments and championships. Accept these invitations and you will begin competing in a tournament. Finishing in first will award you the achievement for said tournament.
The simplest way to win these is to set the frames to the minimum (3) and then after completing each match saving your game. If you happen to lose a match or are about to return to the dashboard and then reload your previous save.
You will unlock most of the matches as you reach the top 16.