Instructions
Use your Slide/Dodge roll/Cartwheel dodge move liberally. Which move you use depends on the character you are using at the time. Most of Vanitas' Lingering Spirit's attacks cannot be blocked, so the only way to avoid taking damage is to dodge them.
Equip lots of potions before starting this fight. The Lingering Spirit will unleash devastating attacks on you if you try to use cure spells, so your best method of healing your character is through potions.
Equip your strongest weapons and make sure you have earned the "Once More" and "Second Chance" skills. These will be invaluable in your battle with the Lingering Spirit.
Lure Vanitas' Lingering Spirit into traps. Do not try to engage him directly, as he will cut you to ribbons. Instead, lay down a series of traps -- Mine Shield, Mine Square or Seeker Mine work best -- and lure the boss into stepping into them. The traps will go off and stun Vanitas' Lingering Spirit momentarily, enabling you to rush in and get a couple good hits.
Play to each character's strength. If you are playing as Terra, stun Vanitas with a trap, and then close in and pound him with your strongest attacks until he recovers. If you are playing as Ven, spam your "Tornado" move when Vanitas is stunned or when there is an opening in his defenses. Aqua should rely specifically on Seeker Mines to both stun and damage Vanitas.