Like any other character's story mode, the final boss is Marie. I've always found Marie to be tough, especially as Double (I don't know Double much as a character, I've said this before). Your best bet to to spam your level 1 blockbuster whenever you get it (or you can save it for a method I'll cover later).
j.MP (Jumping middle punch) is a good multi-hitting move that will do a good amount of damage. You should probably spam this while Marie is in her first form. Also, ignore that Steam notification.
Remember how I said you can save your meter for a later time? Well, now's that time. If you saved your meter, you can use your Level 2 blockbuster when Marie is in her second form. The move will summon a bunch of Ms. Fortune heads that attack whenever you land a hit, adding on a ton of damage. The input for this is QCB + KK (Quarter-circle back plus any two kick inputs).
From there, just spam attacks until Marie is in her third form.
Once Marie is in her third form, use your level 1 blockbuster if your meter is filled (The input again is QCB + PP).
After landing the Level 1 blockbuster, just spam j.MK to lower her health a lot. Once she's dead, you've completed Double's story mode. Click "Next Page" to move on to the story mode for Ms. Fortune.