Mission 35: Collect hearts
Collect more hearts.
Mission 35 is the first of a few missions that require a task to be completed before it becomes available. In this case, all the treasure chests in Mission 29 must be opened. Afterward, talk to Axel in the Grey Area anytime between days 97 to 100, and this mission will become available. If you miss this mission the first time through or are just interested in completing the game, have no fear, as it and the other special missions will open if you finish the game.
You will start in the West Hall, near the entrance to the Undercroft. You'll probably want to go ahead and walk around the area, seeking the many chests and getting rid of the first of the Heartless. There are mostly Shadows here, but there are also a couple of Snowy Crystals as well that will count towards your total. There are any number of paths to take. The West Wing is a dead end, so you may want to clear it out first, and then use the Secret Passage and Undercroft to circle back to the RTC point.
The West Wing holds three Snapper Dogs and more Shadows, as well as a high Unity Badge near the entrance. You'll need to come back with High Jump if this is your first time through. The Secret Passage across the way via the West Hall has Massive Possessors, which won't count towards your total, and Snowy Crystals that will. The Undercroft, however, holds the motherlode: tons of tiny, but regenerating Icy Cubes. This will easily count towards the bulk of your goal. If you have gotten the other Heartless throughout the castle, this will likely finish both the regular and optional mission as well. Once you're satisfied with the number of Heartless you've gotten rid of, you can RTC.
Bring plenty of Fire casts as all the emblem heartless are weak against it (using Doublecast panels or higher is definitely essential) and some Hi-Ethers for restoring casts if needed. For dealing with the Icy Cubes in the Undercroft, take down as many as you can with Fire while taking damage and then quickly finish the rest off using a Limit Break. Overall not very difficult as long as you know where to go and are capable of defeating the emblem heartless quickly enough.