She went there because she was running away from Sandor and the fact that all her family had died, and the only refuge she had was the coin Jaqen gave her. She killed people on her list multiple times before she ever got to the FM, and conceivably, she'd have better access to kill the rest if she went North, fomented a Stark rebellion, and became part of the Northern cause to end Lannister rule. I'm not sure book or show Arya went to the FM with the idea she'd become an assassin, she just wanted sanctuary.