Once you make a save file after completing the Final Episode you can reload any save with a "*", a starred completion, and go to the Keyblade Graveyard. Here you can choose to land at the "Badlands" where you examine an orb near the spire to trigger the battle.
BOSS - Vanitas Lingering Spirit
For starters, this battle is insanely difficult. Practically any attack that lands on you will be a one-hit kill, thus Second Chance and Once More are mandatory. Though Vanitas has very low HP he has high defense and won't take much damage from magic. Don't ever use magical healing abilities, ie Cure, Cura, Curaga. This allows Vanitas to mimic you and heal himself. Instead, equip Potions and Hi-Potions because, regardless of the HP they give back, you'll likely end up at 1 HP remaining anyway and Vanitas can't copy this. Use the standard dodging techniques and not their offensive counters because they provide a longer period of invulnerability and you never want to block Vanitas's attacks, rather simply dodge them.
Few attacks are effective here but Ventus's Tornado can deal some good damage. Aqua and Terra need to rely mainly on Mine Shield, Mine Square, and Seeker Mine (for Aqua). Set a trap and then put it between you and Vanitas so that he falls into them and takes some damage. It will take practice to learn when you can use a potion but generally a good time is immediately after a dodge. Every now and then you can actually get in and hit Vanitas with the Keyblade but it's best to keep distance and drop Mines and dodge.
BOSS - Mysterious Figure
Like the previous secret boss the enemy will deal enough damage to leave you at 1 HP everytime. Fortunately, you can use healing spells in this fight so equip them. Either equip lightning fast commands or simply load all the slots up with Curaga and rely on the Shotlock, which will be the heart of your attacks anyway. The best times to heal are after you have been knocked up in the air or back away from the Mysterious Figure. You will be countered basically any time you try to melee so strike from distance or with those quick commands like Sonic Blade. There isn't much to the fight but it is hard, so stick to dodging and waiting for the right time to use a Shotlock.