You get these achievements by defeating their respective bosses. Ideally you get all of them just by playing the game, but if you're unlucky, here's a list of where you can find each boss. You can use the Contract item to change the boss of a stage & you can quit and restart the game to reroll more than once (for chapter 6 bosses).
Also, beating a boss in the Abyss will not unlock the achievement.
Elf Swamp: Draserpent | Mudtain
Crystal Valley: Condemner | Crystalforger
Moon Grove: Ursa | Blood Chief
Wudie's Glacier: North Wind | Sin Eater

Wild Forest: Queen Beelica | Warden
Sand Field: Crossbow Brothers | Despairer
Ancient Highland: Bonfire Knight | Aerolistar
Bogle's Garden: Batragon | Heart Colossus

Red Hill: Iron Champion | Evision
Northern Land: Frostfang | Winter General
Sage Relic: Bladehand | Watcher
Bogle's Garden: Batragon | Heart Colossus
Jade Forest: Azure Centaur | Deaterfly

Elf Swamp: Draserpent | Mudtain
Crystal Valley: Condemner | Crystalforger
Moon Grove: Ursa | Blood Chief
Dune Sea: Eternal Guard | Karma
Hiden King's Land: Oracle | Red Lotus | Sigma

Wild Forest: Queen Beelica | Warden
Northern Land: Frostfang | Winter General
Ancient Highland: Aerolister | Bonfire Knight
Bogle's Garden: Batragon | Heart Colossus
Jade Forest: Azure Centaur | Deaterfly
God's Domain: Hero King | Bliss Bird | Punisher

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Queen Beelica
Defeat Queen Beelica


Warden
Defeat Warden


Iron Champion
Defeat Iron Champion


Evision
Defeat Evision


Draserpent
Defeat Draserpent


Mudtain
Defeat Mudtain


Frostfang
Defeat Frostfang


Winter General
Defeat Winter General


Crossbow Brothers
Defeat Crossbow Brothers


Despairer
Defeat Despairer


Condemner
Defeat Condemner


Crystalforger
Defeat Crystalforger


Bladehand
Defeat Bladehand


Watcher
Defeat Watcher


Ursa
Defeat Ursa


Blood Chief
Defeat Blood Chief


Bonfire Knight
Defeat Bonfire Knight


Aerolistar
Defeat Aerolistar


North Wind
Defeat North Wind


Sin Eater
Defeat Sin Eater


Eternal Guard
Defeat Eternal Guard


Karma
Defeat Karma


Batragon
Defeat Batragon


Heart Colossus
Defeat Heart Colossus


Deaterfly
Defeat Deaterfly


Azure Centaur
Defeat Azure Centaur


Oracle
Defeat Oracle


Yama
Defeat Yama
Known as Red Lotus in-game.

Sigma
Defeat Sigma


Shining Starry
Defeat the gods with the help of the crowd


Dawn Comes
Defeat the gods with the power of Venus


There's a lack of tips for the boss fights, but I will say that if your team isn't really good when fighting god, you should go for 'Dawn Comes' first (accept authority).

Cavalry Legion
More than 5 Ninelie Cavalries in your team when you clear the Chapter
This is up to luck, but you can maximize your odds by buying the followers for the reptile valley early & always having a spare prayer stone.

Colossus Legion
More than 3 Colossi in your team when you clear the Chapter



Remember that Travel difficulty lets you choose your starting 4 members, so you only need to find one more for the achievement. I.e. you can start with 4 warriors and then hope you find another one or guarantee it by bringing a Draft item.

Warrior Warband
More than 5 Warriors in your team when you clear the Chapter


Guard Warband
More than 5 Guards in your team when you clear the Chapter


Scout Warband
More than 5 Scouts in your team when you clear the Chapter


Hunter Warband
More than 5 Hunters in your team when you clear the Chapter


Priest Warband
More than 5 Priests in your team when you clear the Chapter


Legendary Journey
No character is killed when you clear the Chapter


Financial Freedom
You have more than 2000 Coins when you clear the Chapter
I highly recommend starting with the Golden Hand treasure (50% more coins on battle win) if you're going for this. The Kodiak outfit lets you choose a treasure to start with or you can use Wind Chimes to reroll the first treasure altar. Note: Restarting the game affects the reroll selection but not the initial selection.

Beast Break Out
Clear Chapter VI with Ninelie


Sword Spread Out
Clear Chapter VI with Bronion


Break the Shackles
Clear Chapter VI with Evernight

Bringing the treasure in the achievement picture to its respective boss will start a fight where you don't fight the boss directly and instead do some different objective. Note: This does not work in chapter 7 or Abyss, based on my experience. Edit: Chapter 7 fixed in v1.0.3

The proper way to get these is to wear Kodiak outfit to start with the treasure you need and bring a Contract to switch the boss if necessary. However, if you want to wear a different outfit (like Perseus), you can use Wind Chimes to reroll the treasure altar and restart the game as necessary. Restarting the game affects the reroll selection but not the initial selection.

Additionally, you can lock treasures to not appear to make this process slightly more consistent.
Frost Wolf Legion, Ninelie Hero, Bronion Crown, & Evernight Training require treasures that need to be unlocked from the Lodge.

Beelica Hero
Save Queen Beelica from Wudie control


Champion Road
Get Iron Champion drunk and have a good sleep


Back to Homeland
Guide the soul of Crystalforger to rest


Watch the Dawn
Let Watchers regain their consciousness


Frost Wolf Legion
Help Frostfang guard Wolf Priestess


North Wind Dragon
Go to the North to scare away the Acient Dragon named North Wind


Ninelie Hero
Help Eternal Guard retrieve his memory


Bronion Crown
Attack Wudie Camp with Azure Centaur together


Evernight Training
Pass the special test of Karma

The description says "Endless Mode", but it's actually Desperate Journey only.
Source: https://gtm.steamproxy.vip/app/1937750/discussions/0/3713810544750778686/
A few tips:
- Boss stages in 1-50 are all the bosses excluding chapter 7 bosses, then it repeats for 51-100 but with harder modifiers.
- The harder modifiers will include 30% crit resist which means you'll need 130% crit chance to always crit.
- Equipment that gives bonus crit chance, crit damage, dodge chance is good since you can get way more of them compared to journey mode.
- Death god bloodline is extremely good since you can't revive units otherwise. Although it is a rare event & prayer stones don't appear often.
- Ninelie is the easiest clan because their scouts have some busted traits. This is even more true since promoting a unit to cavalry allows you to draft that unit again.

A summary of my build that was first/second turn killing bosses after level 70.
All units promoted to cavalry, but I'm referring to their pre-cavalry names.
- 3 scouts all with Synergy: 1 Basat to backstab every single regular enemy & 2 Golina to stun bosses for 2 turns. (Still have to be careful about some abilities that activate while stunned.) With 125% crit chance by buying scout equipment at every opportunity + extra damage traits & dumping every strength upgrade into Basat, she was doing about 60 damage per action.
- 5 priests: 2 Montan, 2 Snow Song, 1 Leiya to gather aggro with the summons.
- 1 guard: 1 Mousse with Horn, which only got used on Sigma.
- 1 hunter: 1 Moonlie, made the early encounters easier & never needed to swap her off.

Endless Bronze
Clear 5 Stages in Endless Mode with any Clan


Endless Siver
Clear 10 Stages in Endless Mode with any Clan


Endless Gold
Clear 20 Stages in Endless Mode with any Clan


Endless Platinum
Clear 30 Stages in Endless Mode with any Clan


Endless Diamond
Clear 50 Stages in Endless Mode with any Clan


Endless Master
Clear 80 Stages in Endless Mode with any Clan


Endless Challenger
Clear 100 Stages in Endless Mode with any Clan

Legendary Strike
Deal 100+ Damages in one Attack
There's an elite battle in Bogle's Garden where you guide a dive bombing bird who does way more than 100 damage. I don't know how else to get this achievement.

Perfect Kill
Clear the BOSS stage without any damage
You should get this by playing normally, but a guaranteed way is to fight Queen Beelica with multiple hunters (as Ninelie).

Lonely Hero
Clear the BOSS stage with only one character
There're multiple ways to go about this, but Iron Thorn Stone or Gold Stone with a lot of armor works well.

The Blitzkrieg
Clear the stage in one turn
The Flame Meteor treasure will actually insta-clear some early encounters. Regardless, you'll eventually get this by playing normally.

Penta Kill !
Kill more than 5 enemies in one attack
You should also get this through normal play. Tiger Colossus is best for this, especially since you can get it in stage 1 where you can easily bunch up weak enemies.

Penta Revive !
Revive more than 5 characters at a time
Keyword is more. The achievement name is a misnomer since you have to revive 6 at minimum. Best way to get it: play Conquest difficulty, find a Sacred River, then sack your whole party and revive them.

Traveler Diary
Unlock all the characters in Illustrated Book
Character info is filled when she enters your party. If you're missing anyone, spam random on the party select screen for Travel difficulty and start the journey to get their unlock.

The Dream Begins
Start your Journey at the first time
Why this isn't listed as the first achievement is beyond me.

Not One Less
Unlock all the spots in single Journey
This means clicking on every spot to get rid of the exclamation mark.

Play the game and spend your resources as much as possible.

Wishing Expert
Consume 200 Prayer Stones in total
If you have to grind Prayer Stone usage (like me), fastest way seems to be starting a Bronion Travel journey, doing 2 encounters, forging a colossus, & repeat.

Blood Crystal Palace
Consume 500 Blood Crystals in total
To dump Blood Crystals, spend extras on rerolling a level-up trait; you can keep a character with an unchosen level up until you arrive at the last boss of the chapter.

Million Households
Consume 10000 Coins in total