Man, that dang "Instrument" is evil! The instrument killed Sona's adopted mother simply b/c Sona loved the woman more than it/music. It committed the murder out of jealousy, wanting all of Sona's attention and love for itself.
The reason Sona wants to join the League is b/c she needs to learn complete control over her Instrument, so that what happened to her adopted mother never happens again. By joining the League, she will expand her skills, and makes it clear to the Instrument that she is the MASTER, not it.
Also, she wants to be famous across the land. Since she lived a lonely life, she wants people to know of her through her music. For that, she needs a bigger stage, and there's no bigger one than the League of Legends.