The Wheel of Harma Trials become available after you finish the events at Angri-La in Act 2. To start the Trials, go to The Field of Discipline and speak to the High Lama. There are five overall trials you need to complete, and for the achievement you do not need to complete the battles within enough turns to achieve all rewards, although many rewards are outfits for your characters which you will need for another achievement.
To unlock the last of the trials you have to do wait until you go to Havens Above in Act 3/Post Game. I would recommend waiting until level 99 along with having the best gear you can before attempting the fifth trial.
Here is the strategy I used for each trial as shown by the FAQ guide on the game by Azmuch. I also had Catholicon Rings +3 for every character, though this was a little unneeded.
Trial 1:*** Spoiler - click to reveal ***
Battle 1:
Participants: Jade.
Jade uses Multithrust each turn.
Battle 2:
Participants: Hero and Rab
Hero uses Quadraslash each turn.
Rab uses Pearly Gates each turn.
Trial 2:*** Spoiler - click to reveal ***
Battle 1:
Participants: Erik
Dual-wielding boomerangs, uses Divide followed by Double Down.
Battle 2:
Participants: Jade
Use Pink Tornado each turn.
Battle 3:
Participants: Hero and Sylvando
Hero uses Quadraslash each turn.
Sylvando uses Gold Rush each turn.
Trial 3:*** Spoiler - click to reveal ***
Battle 1:
Participants: Sylvando
Use Gold Rush each turn.
Battle 2:
Participants: Hero
Use Gigagash if he has learned it. Otherwise use Quadraslash.
Battle 3:
Participants: Rab and Serena
Rab uses Pearly Gates.
Serena casts Kacrack for ice-elemental damage to all enemies.
Battle 4:
Participants: Erik and the secret character
"?" attacks Badonis with Parallax.
Erik, dual-wielding boomerangs, uses Divide followed by Double Down targeting Badonis.
Trial 4:*** Spoiler - click to reveal ***
Battle 1:
Participants: Jade, Rab, and Serena
Jade uses Pink Typhoon.
Rab casts Thwack.
Serena casts Kathwack.
Battle 2:
Participants: Erik and the secret character
"?" attacks Clangoustine with Unbridled Blade.
Erik uses Critical Claim on the Hardy Hands.
Battle 3:
Participants: Veronica and Sylvando
Veronica casts Magic Burst.
Sylvando uses Ladies First to pass his turn to Veronica, if he goes first.
Battle 4:
Participants: Hero
Attack with Sword Dance
Again, going into this trial without being fully prepped and at level 99 will be a challenge. After every few turns or anytime you kill a monster, "back up" will show up, and if not done properly can send you into a infinite loop. I would recommend everyone being pepped before going into the fight with Veronica and Serena being mandatory in being pepped.
Trial 5:*** Spoiler - click to reveal ***
Battle 1:
Participants: Rab and the secret character
"?" attacks Shadeshifter with Unbridled Blade.
Rab casts Kacrackle.
This was the hardest of the battle for me, I made sure rab had as much magic might going into this fight and casted Infinite Wisdom on the first turn and defended with "?" until the next turn. If back up shows up, just flee and try again.
Battle 2:
Participants: Hero and Jade
Hero uses Sword Dance on Abyssal Octopot and Gigagash when backup appears.
Jade uses Multifeet on Abyssal Octopot and Pink Tornado when backup appears.
Make sure you kill Octopot before the backup dies or he may bring them back to life.
Battle 3:
Participants: Erik and Sylvando
Erik, dual-wielding boomerangs, uses Divide followed by Double Down targeting Pandora's Big Bad Box.
Sylvando casts Oomphle on Erik and uses Gold Rush when backup appears
This one is pretty easy, don't skip out on Oomphle.
Battle 4:
Participants: Serena and Vericona
Whoever takes the first turn uses the Mighty Magic Burst Pep Power
This should one shot them if you are at level 99 with good equipment.