The Burst Rush skill is unlocked during Mission 3. Basically, you move the left stick and press B to do a dodge, then press X to follow up an attack. The game explains it a little better.
You'll probably get this just by using the technique against enemies. If not, try it out in a challenge room. I got it by attempting one of the nurse's challenges, which can be selected from the Mission Select (Mondo Girls 2). Challenges are unlocked by finding the nurse in story missions.
Here are six places early in the game to unlock nurse challenges. Talk to the nurse with A after finding her at each spot.
Chapter 3 - Up the stairs, at the beginning, before going in the side room.
Chapter 3 - In the main area, stand on the heart symbol a few seconds for then go up the stairs.
Chapter 3 - In a hall with several cat statues on both sides. Break them all.
Chapter 4 - After going through the first painting, break pots on the left and use the Drill" add-on weapon to break an obviously-breakable wall.
Chapter 4 - Right after, go upstairs, break a pot, press a green panel. Go back downstairs, through two doors, and break all the pots in the room.
Chapter 4 - Toward the end, after the checkered floor room and room with many benches, back in the main area, go up on a balcony overlooking the next room. Do this before going in the last picture in the center of the wall below.