Edit: For some reason, this unlocked for me without me having to get on stage and perform a poem of my own. However, I have been told you need to perform yourself and then get Olivia to perform, and she will copy your poem, unlocking the achievement.
During Year 2, enter the Blue Casket and head to the back of the bar. While speaking to Olivia, you can ask her to perform a poem on stage. The achievement popped for me on the first poem, but you can ask her to perform several times before she begins to repeat them.