There are six Plane of Justice trials. Completing any one trial is enough for planar progression flagging. Completing all six unlocks The Seventh Hammer. Seventh Hammer is not required for planar progression. Full raid order is on Planes of Power Progression.
Access
Find Mavuin in Plane of Justice. Hail him and say information.
Complete a trial and loot the Mark. Hail one of The Tribunal in the trial room and say mavuin sent me. Hail Mavuin again.
Starting and leaving
Each Tribunal starts one trial. Hail them, say prove, then prepared. When you are ready, say ready to begin the trial of [name]. The start names are Execution, Flame, Lashing, Stoning, Hanging, and Torture.

If that trial is already running, the Tribunal tells you to wait.
To leave, hail Agent of The Tribunal and say return. If the trial is still underway, the Agent will not port you out. Fail lockout is 60 seconds. Success lockout is 30 minutes.
Trial of Execution
Generally the easiest trial. Four waves of four mobs spawn. Kill one mob at a time and keep DPS up. A hooded executioner walks toward the victim / guillotines during each wave. If he reaches them, the trial fails.
The executioner only dies when all accompanying nemeses are dead.
After the fourth wave, Prime Executioner Vathoch spawns. Kill him and loot Mark of Execution.

<Prime Executioner Vathoch> - CH Chain: 4sec, 6sec slowed, Hits 720, AE Rampage, Slowable (90% mit), Casts: Armor Break (PBAE, -265 AC / -354 cloth, Unresistable, 2 min), 38.5k HP
Trial of Flame
Do not let the fire creatures reach the center of the room or the trial fails. Four waves, then Punisher of Flame. Kill him and loot Mark of Flame.

<Punisher of Flame> - CH Chain: 4sec, 6sec slowed, Hits 746, AE Rampage, Slowable (90% mit), Fire Immune (FR 1000), Casts: Rain of Molten Lava (PBAE, 300 DD, -100 Fire, 12s recast) plus wizard nukes/roots, 32.3k HP
Trial of Lashing
Four waves, then a boss. Each of the first four waves has four targetable mobs and one untargetable a scourge of honor. Kill the four targetable mobs. When they die, a flickering spirit spawns in each of the three hallways. Kill those, then the scourge despawns and the next wave can start.
After the fourth wave, Lashman Azakal spawns. Kill him and loot Mark of Lashing.
<Lashman Azakal> - CH Chain: 4sec, 6sec slowed, Hits 720, AE Rampage, Slowable (90% mit), Casts: Thunderclap (PBAE, 200 DD + 2s stun, Magic, 30s recast), 36.3k HP
Trial of Stone
Start this one as trial of stoning. Mobs spawn from the doors. Four waves of four mobs. Spectres can be mezzed. Untargetable archers will not stop until nothing else stands — do not let the prisoners take more than they can bear. Those shooters despawn when the four targetable mobs die.
After the fourth wave, Yurae Zhaleem spawns. Kill him and loot Mark of Stone.
<Yurae Zhaleem> - CH Chain: 4sec, 6sec slowed, Hits 720, AE Rampage, Slowable (90% mit), SK, Casts: Gravel Rain (PBAE, 250 DD + 55% snare, -300 Magic, curse, 30 min), 42.4k HP
Trial of Suffocation
Start this one as trial of hanging. Waves spawn a few at a time. Kill spirit of suffocation mobs first or their victims perish and the trial fails.
After a few waves, Gallows Master Teion spawns instead of wraith of agony. Kill him and loot Mark of Suffocation. He is an Enchanter (Gravity Flux, mez, tash).
<Gallows Master Teion> - CH Chain: 4sec, 6sec slowed, Hits 720, AE Rampage, Slowable (90% mit), Enchanter, Casts: Gravity Flux and mez/tash/root, 32.3k HP
Trial of Torture
Four waves of four mobs, two per side. After each wave a wraith of agony spawns in the middle room or the zone-in cubby. Kill it quickly or the trial fails. There is a short wait between waves.
On the fourth wave, Punisher Veshtaq spawns instead of wraith of agony. Kill him and loot Mark of Torture. He is Unslowable.
<Punisher Veshtaq> - CH Chain: 3sec, Hits 720, AE Rampage, Immune to Slow, SK, Casts: Pain and Suffering (PBAE, 100 DD, Unresistable, 15s recast), 42.4k HP
The Seventh Hammer
Do all six trials and loot the Mark from each. Say knowledge to one of The Tribunal in the trial area to receive The Mark of Justice. Hold The Mark of Justice on your cursor and click a blue portal behind a Tribunal to zone up (U can be needed). The Mark of Justice is not a key-ring item. One person per group can port the group.

The Seventh Hammer is Unslowable, AE Rampages, hits 1240, and has 1.08M HP. Tremor of Judgment and Verdict of Eternity are cast by an invisible Tribunal on a ~3 minute cadence.
<The Seventh Hammer> - CH Chain: 2sec, Hits 1240, AE Rampage, Immune to Slow, Dual Wield, Procs: Justice (Single, 800 DD, Unresistable), Casts: Tremor of Judgment (PBAE, 800 DD, -300 Magic), Verdict of Eternity (PBAE, -100 mana/tick + 200 HP/tick, Unresistable, 9 curse), 1.08M HP