Missable!
This achievement is missable if the first time you encounter QuAsSy is also the time you help Slash defeat her. Helping Slash defeat QuAsSy means you will help him on subsequent playthroughs and so you will have to reset your progress in order to get this achievement if you didn't get it first.
Thanks to Gensui Akainu for suffering.
Location
This is the easiest boss to find as it is always in the same place: atop the Space Elevator. When you make your way to the steps at the top of the level, QuAsSy will talk to you and then the fight will begin.
Fight
The hardest part of this fight, for me, was figuring out what to do and once I had done so, this boss was easy. In order to damage it, you have to destroy all the lasers it uses by hitting them at their base. When all the lasers are destroyed, QuAsSy will open itself up for you to get one hit in. It takes four hits to kill it and each time the lasers come back, there is one more than last time. As well as the lasers, which are fired in a random pattern, QuAsSy spawns multiple enemies to hassle you. They are more of a mild inconvenience and can be totally ignored if you have enough health and defense on your build, along with a powerful, upgraded weapon.
When you have hit her four times, you will get some dialogue, a cut scene and this achievement.