You can do Side Quests and Errands to boost your party if necessary, but except for those only do the quest in the lists to trigger the achievements.
All classes are unlocked through side quests.

Chapter 1

• Monster Infestation
  
• Ancient Ruins
  
• Inspection
  
• Short Supply
  
• Another Pledge
  
• Royal Summon

Chapter 2

• Find the old Knight
  
• Red Reception
  
• White Wandering
  
• Black Brawl
  
• The Magi's Apprentice
  
• Scout the Scout

Chapter 3

• Trials of Valthiria I
  
• Trials of Valthiria II
  
• Trials of Valthiria III
  
• Trials of Valthiria IV
  
• Grave Danger

Chapter 4

• Shermantree I
  
• Shermantree II
  
• Shermantree III
  
• Shermantree IV
  
• Trading Ring
  
• Kawiyan Gauntlet I
  
• Kawiyan Gauntlet II
  
• Kawiyan Gauntlet III
  
• Kawiyan Gauntlet IV
  
• Homecoming
  
• Escaped Beak
  
• Northern Scimmage

Now start the final quest War of Five Queendoms

Reload your save and do the following quests

• Infamous Howl
  
• Blue Monsters
  
• Truant Twins
  
• Mining Fiesta
  
• Search and Rescue
  
• No Gem no Life
  
• Troll Guard
  
• Fearless Boy
  
• Critical Reinforcement
  
• Critical Desolation
  
• Critical Rescue
  
• Spelunked

the final quest should have changed to The Covenant. Finish it to trigger the achievement.

Reload your save for the last round.
Just complete all quests that are left , finish all achievements still open and do all principle tasks (now there should be just 2 achievements left: Valthirian Peace and The Completionist) before starting the final quest The Covenant.
Unfortunately the Completionist isn't working atm.