The First thing to know about Amaterasu is her unrivaled mobility and utilities for attack and defense make her a threat smack dab in the middle of a fight. I would recommend building only Power, Penetration, and Attack Speed on her. Crit will not help you on Amaterasu even though that is what some would lead you to believe. also it is imperative to assign her stance change as your first skill. this skill does not only affect her but your teammates as well, her mobility stance could help a teammate in trouble get away from a possible champ death. I would not build up her mirror skill until you have at least one point in Glorious Charge. she has a combo that eats health with her power stance, where you charge the mirror, throw it, charge behind the mirror, and spam basic attack to kill the opponent. very effective very fast. and when you get her Ult, and add it on to the end of this combo, very few escape with their health intact. as far as relics, I would recommend Horrific Emblem first to set up those early ganks, great feeding or eating tool and also teleport glyph, Amaterasu doesn't really have too much issue getting back into the fight due to her high mobility, but if something should happen in lane mid game where you're absolutely needed, it's imperative to have this as a second relic. when setting up ganks as Amaterasu, it is important your teammates know whats about to happen or else they'll all go in blind and you'll fail. communicate, practice in sync combos with your newfound favorite champ and my overall favorite champ, Amaterasu!
That is all the advice I have, thank you. I can try to answer any specific questions should there be any!