The game is advertised as free but I noticed that the achievement points aren't credited to your gamerscore. Does it unlock when you buy anything or just the Hawaiian Adventure expansion?

Are you sure you're getting the achievements and not the awards?

You have to be signed into your gamertag to unlock achievements since it's a GFWL game. You'll know when you get one because when they unlock it makes the trademark unlock sound and points are associated with it. What you're seeing are probably just the awards that pop up in game that level you up.