After completing a world, an Unversed emblem will appear in a specific area of that world depending on what character you're playing as. When you interact with it, a special minigame will start. You'll need to complete all of these minigames/battle missions with the highest rank (3 stars) for this trophy. You only need to complete them with one character for this trophy. Keep in mind that all three characters need to complete the Unversed missions to get the report entries, meaning you do not need to obtain a 3-star rating with the other two characters on the missions in Enchanted Dominion, Deep Space, and Olympus Coliseum.
Below is the list of locations and strategies for each character. You can keep track of what Unversed battle missions you've done in the game from the pause menu by going to Reports, Game Records, and then choosing Unversed Missions.
FLAME BOX
In this battle mission, a chest-like Unversed will run around spitting fireballs. You need to destroy at least 30 of these fireballs in order to get 3 stars and pass the battle mission.
Location (T/V/A): Enchanted Dominion - Waterside
Reward: Firaga
Strategy: It's the same for all three characters. Just go in with 3-4 Magnega commands, and use them as Flame Box shoots out fireballs. Magnega will automatically destroy them for you. Just keep destroying fireballs until you hit 30.
LONE RUNNER
In this battle mission, a bunch of boot-like Unversed will run around the room. You'll need to destroy at least 30 of them to get 3 stars and pass the battle mission.
Location (T/A): Castle of Dreams - Ballroom
Location (V): Castle of Dreams - Wardrobe Room
Reward: Illusion-L
Strategy (V/A): With Ven and Aqua, you'll want to equip one or two Mega Flares, then spend the rest of the slots on Magnega, and one surge command. You don't want just Mega Flare, as they don't recharge fast enough to spam, and you won't be able to kill 30. Anyway, use your Mega Flares on the first couple of groups, then run/superglide to each group as they spawn and use a Magnega. Use your surge command on any single lone runner's that get away from the Magnega.
Strategy (T): The strategy is basically the same with Terra. The only difference is that you'll want to use the maxed out Ven D-Link. It gives you haste, which helps balancing out Terra's ridiculously slow attacks.
VITALITY VIAL
In this battle mission, the entire area will be filled with poison, and your health will constantly drain. When you hit the Vitality Vial, it heals you, but only with regular attacks. Commands don't work. In order to get the 3 star rank, you have to survive for more than 2 minutes.
Location (T/A): Dwarf Woodlands - Underground Waterway
Location (V): Dwarf Woodlands - The Mine
Reward: Illusion-V
Strategy (T/A): With Terra and Aqua, you'll want to use the maxed out Ven D-Link. It will give you Haste, making it much easier to hit the Vial and recover your health. Hitting the Vial with commands doesn't heal you, so just go in with all Curaga commands. You can use them if you're having trouble reaching the Vial before your health runs out.
Strategy (T): Another strategy for Terra is to use the Demolition Finisher. Just go in with 8 Curagas for emergencies and hit the Vitality Vial to build up your finisher gauge. Afterwards activate Demolition, this finisher takes a long while to complete and you'll be completely immune to the poison during that time. Equipping the Royal Radiance keyblade will quickly fill the finisher bar. See Silver Darkness' Will on how to obtain it.
Strategy (A): Aqua can also use her Teleport Spike finisher to stay immune against the poison. The finisher will knock the Vitality Vial down, so afterwards rush towards it, fill the finisher gauge, and repeat.
Strategy (V): Since you don't have Haste when using Ven, you just have to rely on your skill when trying to hit the Vial. Once again, you'll want all Curaga commands. Using Superglide to chase the Vial is very helpful, make sure you've got it equipped.
Strategy (V): Just like with Terra, you can use the Stratosphere finisher to make this even easier, as it grands immunity against the poison. As an added bonus the finisher will knock the Vitality Vial down when pressing the correct buttons 4 times in a row. This makes it very easy to chain finishers together and you won't even need a Curaga. Remember to use the Royal Radiance keyblade to fill the finisher bar more quickly.
BELLY BALOON
In this battle mission, a giant fat Unversed will appear in the area. The objective is to decrease the gas gauge to zero within 1 minute. To decrease the gauge, you'll have to hit its back. When you hit its belly, the gas gauge will refill.
Location (T/V): Radiant Garden - Outer Gardens
Location (A): Radiant Garden - Central Plaza
Reward: Illusion-B
Strategy (T/V/A): Equip all surge commands. When it turns around, start mashing surge commands. It's a fairly simple mission, you just need to time your attacks. As Aqua, keep in mind to unequip the action command Teleport to make it easier.
RINGER
In this battle mission, a jar-like Unversed will split into multiple parts. You'll need to find the real one and destroy it. You have a total of 2 minutes to get 40 points. Every real Ringer gives 1 point for every remaining Ringer. Every fake one gives 1 point, but also reduces the time limit by 5 seconds.
Location (T/V/A): Disney Town - Raceway
Reward: Illusion-R
Strategy (T/V/A): This mission is the most skill based. You have to watch and see which Ringer is the real one, and kill it. You can pause the game when the fake Ringers appear and they should appear semi-transparent. That way you'll know which one is the real Ringer. Afterwards keep an eye on it, lock on, and hit it.
GLUTTONOUS GOO
In this battle mission, a giant goo Unversed drops orbs for you to collect every time you hit it. Whenever it hits you, you'll drop orbs you've collected. There's a total of 100 orbs, and you'll need to get at least 90 to get a 3 star rating.
Location (T/V/A): Deep Space - Launch Deck
Reward: Stun Block (T/V) or Stop Barrier (A)
Strategy (T/V/A): You'll primarily want to block its attacks with
while standing still, and counter with
to get orbs (by counter and block, be sure to have Counter command attached to Block under Action Commands). If you try to just attack it, it will just counterattack and you'll lose orbs. The goo won't drop orbs if you attack it with commands, so you just have to block and counter until you've gotten more than 90. Note that the Gluttonous Goo will sometimes counter after your counter. Having Payback Fang/Strike/Barrier will enable you to attack after the counter. It seems that jumping after a counter will net you more orbs (remember Aqua has a double jump later in the game). That's because the Goo floats and is taller than your character, which means it'll attract most of the orbs. It also has some kind of Treasure Magnet ability. You'll also want to have the Treasure Magnet ability maxed out for this mission or else it will prove to be a massive pain. As Aqua, keep in mind to unequip the action command Teleport to make things easier. Once you've gotten enough orbs, just run away until time runs out.
ELEMENT CLUSTER
In this battle mission, a totem-like Unversed will switch between three elements: Fire, Ice, and Thunder. You'll need to attack the Cluster with its current element to earn points. You can tell which element it currently is by looking at the middle totem. If it's the red one it's fire, blue it's ice, and yellow it's thunder. You'll need more than 70 points for the 3 star rating.
Location (T/V/A): Neverland - Indian Camp
Reward: Voltage Stack
Strategy (V): The real secret to this mission is using the special elemental movement commands. For Ven, equip Fire Glide, Thunder Roll, and Ice Slide, and use them at the appropriate time to rack up ridiculous amounts of points.
Strategy (A): Just like Ven, you'll need to rely on your elemental movement commands. Equip Firewheel and Ice Slide, and rack up the points. For the thunder section, you can equip normal thunder commands.
Strategy (T): Terra is the hardest to do this with, as he doesn't have an elemental dodge, but it's still not too difficult. Just equip 4 Fire Surge commands, 4 Thunder Surge commands, and Ice Slide.
JELLYSHADE
In this battle mission, you get mobbed by little jelly Unversed. The objective is to defeat 300 within 30 seconds.
Location (T/V/A): Olympus Coliseum - Town Near Thebes
Reward: Stopga
Strategy (T/V/A): This probably sounds like a hard mission, but it's a bit of a joke. Just go in with 3-4 Magnega commands. Use the Magnega commands. You did it.
FLOATING FLORA
In this battle mission, there are 3 plant-like Unversed for you to attack. The objective is to get more than 350 orb. They drop orbs when you hit them in the air. They automatically launch themselves after a couple of hits.
Location (T/V/A): Keyblade Graveyard - Seat of War
Reward: Illusion-F
Strategy (T/V): The easiest thing to do, is to get some Daisy Sorbet Ice Cream, and immediately use it to enter the Rhythm Mixer Command Style. Once you've used the finisher, do it again. Make sure to fill as many command slots with Ice Cream as you can. It's also recommended that you save your game before attempting this, just in case you don't have enough Ice Cream. Just like with the Vitality Vial, use the Royal Radiance to fill the finisher gauge more quickly.
Strategy (A): With Aqua, you'll want to use the Spellweaver Command Style. You can either use the Fabracadabra Ice Cream, or just go into it naturally. Either way, it's incredibly easy go get enough orbs using the finisher.
I take no credit for this solution, Its taken from PSN Profiles, it's the guide I used and it works fantastic, getting the achievement without much effort. I have just tweaked it to make it xbox friendly.