Once you've selected Peacock from the list of characters, you will be introduced to her and given a story arch based around her. I will not show any of the story contents, save for the fights and the ending.
The first opponent you will face is Filia. She is no problem, all I can really say is to spam buttons to ensure you don't get hit.
After defeating Filia, you will move straight on to Ms. Fortune. Again, no real threat, just try and use s.MK (Standing Middle Kick, or B button on controllers) to knock up her head while it's on the floor.
Yet again, you're thrown directly into another battle with no story. This time around, you'll be fighting Parasoul. Watch out for the Napalm shots that her umbrella, Kreig, will fire around the area in order to keep it controlled in her favor. QCB + s.HK (Quarter-circle back and Standing High Kick; may take time for new players to learn how to properly input special moves) will allow you to teleport behind Parasoul and then attack from there.
After a bit of story arch (finally), you'll be up against one of the toughest opponents in Peacock's story : herself. In this mirror match, the opposing Peacock will most likely spam attacks to ensure your movement space is limited. Again, QCB + s.HK is always a good method of attack.
After arriving in the Grand Cathedral, we find two enimes who intend on protecting the Skullgirl. This is a double battle, and it features a very powerful Valentine/Double combo. Valentine is much more powerful than Double is, however. Valentine will take more opportunites to chain long combos instead of being the normal "dumb AI" it usually is. Spam buttons for victory here.
...and down goes Double and Valentine. Here you can see Peacock standing next to an enormous Lenny bomb, who can be thrown onto the field with the Blockbuster input QCF + KK (Quater-circle foward plus any two kick buttons). Also note that using a Blockbuster will require levels (the bar under your character in the top right). Lonesome Lenny only requires 1 level. I felt I needed to put this in for those who don't know about Blockbuster moves.
After beating everyone else, you make your way down deeper into the Grand Cathedral, and finally meet with the Skullgirl herself; Marie. Click "Next Page" for some tips on fighting her as Peacock.