In her first form, I find that comboing j.LP > j.LK (Jumping Low Punch then Jumping Low Kick) will allow a few hits and some quick and low damage.
QCF + s.HP (Quater-circle forward plus Standing High Punch) works well from a distance. It does a reasonable amount of damage and allows for 3 hits if you rapidly press HP.
Peacock's other Level 1 Blockbuster, Argus Agony, is also useful for draining Marie's health. It's best if you use this on her first form, as it will do plenty of damage and knock her back so she can't attack while you're performing it. The input for this is QCF + PP (Quater-circle foward plus any two punch inputs).
Here she is in her second form. As you can see, she is now lower to the ground than she was before. Now more attacks will work on Marie. Just get close button mash for success. If you have any levels left, use them for Argus Agony.
Marie's third and final form is the hardest, even for someone like me who has been playing for quite some time. You can always just use the j.LP > j.LK combo explained in her first form, as she is now in the air.
Make sure to block her shots by moving away from them.
After the vigorous spamming session, you have now beaten Marie! Congratulations on finishing Peacock's story mode. Click "Next page" to move on to the guide for Filia's story mode.