My dream is to become a super strong mahjong player.
Literal
I [of-の] wish [topic-は] very strong mahjong-player [to-become-に-なる] thing is.
とっても is the emphatic casual form of とても ('very'). 麻雀打ち combines 麻雀 ('mahjong') + 打ち ('player, striker,' from 打つ 'to play/hit') — a compound noun for someone who plays. The overall structure is the standard X の願い は Y になることです ('my wish is to become Y').