To unlock Microsoft Gamerscore points, unlock one of the following Achievements.

Achievement	How to unlock
Assemble Your Crew ( 20 points )	Recruit Dean Domino, Christine and Dog.
Cash Out ( 30 points )	Confront Father Elijah in the Sierra Madre's Vault.
Having a Ball ( 20 points )	Complete the Sierra Madre Gala Event.
Safety Deposit Box ( 40 points )	Trap Father Elijah in the Sierra Madre's Vault.
Sierra Souvenir Aficionado ( 30 points )	Collect 500 Sierra Madre Chips.