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All of the info for getting the characters have been found here, and may possibly not be 100% accurate.
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Louie
Go to the Adventurers Guild after visiting the Merchants Guild when you first start out. This character will always be available the first time you try to go on an adventure.

Initial level: 1
Initial equipment: Worn Sword, Leather Armor, Wooden Shield, Old Muffler
Can equip: Sword, Clothes, Breastplate, Bracelet, Shield and Muffler items

Skills
Defend: Block incoming attacks if you stand still or don't attack.
Rotation Slash: Attack everything in a circle around you. Character starts with this skill. Costs 5 SP.
Wave Attack: Character sends a wave out in front of him. Unlocked at level 10. Costs 5 SP.

Charme
Clear Jade Road Floor 15. Have her visit the shop once. Sell at least two items to her.

Initial level: 8
Initial equipment: Rusty Dagger, Scrap Breastplate, Wooden Bracelet, Summer Sandals
Can equip: Dagger, Clothes, Breastplate, Bracelet and Boot items.

Skills
Defend: Dodge attacks by moving towards the direction the attack is coming from.
Dash: Tap the movement key twice to dash in that direction. Can change direction while dashing.
Check Trap: Not sure how it works, but alerts if the chest has a trap if you stand close to it?
Clone: Create clones around the character that are able to attack. Can create up to 6 at a time. Costs 5 SP. Character starts wih this.
Flame Rush: Character rushes forward covered in fire, making you invincible in the process. Costs 5 SP. Unlocked at level 10.
Spider Web: Throw a web in front of you that'll stop/slow enemy movement. Costs 5 SP. Unlocked at level 20.

Caillou
After getting to merchant level 3, Caillou will come in to place an order. Fulfill the order and do business with the character after that to unlock it.

(Incomplete as I haven't played the character at all)
Initial level: 10
Initial equipment: ?
Can equip: Staves, Clothes, Robes and Bracelet items.

Skills
Powered Bomb: Attack in a circle around the character, invincible while using the attack. Costs 5 SP. Character starts with this.
Ice Mine: Place an ice mine on the ground in front of you. If an enemy walks onto it, it'll take low, continuous damage. Costs 5 SP. Character starts with this.
Zapper: Costs 3 SP per second. Character starts with this.
Gemini Force: Attack with missiles all around you. Costs 25 SP. Unlocked at level 25.
Teleport: Teleport randomly around the map. Costs 5 SP. Unlocked at level 30.

Tielle
Meet her during the Morning or Afternoon in the Town Square. Complete Jade Road. Unlocked Amber Garden. Clear Amber Garden (30 Floors) and watch the event. Sell an item to her.

Initial level: 20
Initial equipment: ?
Can equip: Bow, Clothes, Breastplate and Bracelet items.

Skills
Flame Arrow: Fire a fire arrow in front of you. Costs 4 SP. Character starts with this skill.
Seeker Arrow: Fire an arrow that automatically tracks and hits the nearest enemy in the direction fired. Costs 5 SP. Character starts with this.
Holy Rain: Fire an arrow into the sky causing beams of light to hit all around you. Costs 8 SP. Character Starts with this.
Battle Mode!: Paralyse enemies around you for 5-6 seconds. Costs 5 SP. Unlocked at level 25.

Nagi
After day 9, she will randomly start appearing in dungeons. After meeting her 4 times during dungeons, wait for her to visit the shop. Sell to her several times (unknown how many, may be random).

Initial level: 15
Initial equipment: ?
Can equip: Spear, Clothes, Breastplate, Armor, Bracelet, Shield and Necklace items.

Skills
Defend: Can defend against long range attacks if standing still or not attacking. Does not work against bosses.
Flame Rush: Same as Charme.
100 Thrusts: Low power attack that hits several times. Costs 8 SP. Character starts with this.
Zan yen Monthly(?): Attack all enemies around you. Costs 10 SP. Unlocked at level 20.
Wave Attack: Fire a wave at the enemy(?). Costs 5 SP. No idea when it's unlocked.

Elan
Have Charme unlocked. Go to the Pub at night and watch the event. Go to the Merchants Guild during the morning for another event. Have him visit the store once. Sell at least 2 items to him.

Initial level: 15
Initial equipment: ?
Can equip: Gloves, Clothes, Robes, Bracelets.

Skills
Dash: Double tap direction key to dash. Can't change direction while dashing.
Flame Rush: Same as Charme and Nagi.
Flurry: Quick successive punches against all enemies around the character. Costs 20 SP. Character starts with this.
Berserk: Ups Strength, Defense and Magic Defense for 10 seconds. Cannot use items while in this state. Costs 5 SP. Unlocked at level 20.

Grif
One of the harder cahracters to get. Can only be gotten in Endless Mode after paying your debt. Complete Amber Garden. Meet Tielle at the Market and initiate the "I forgot to look for my Sister" event (third or fourth time you meet her at the market). Have Tielle receive a letter from her sister (open the store after the former event to initiate this event). Meet Grif at the Square at Night twice. Have Grif visit the store and give you a warning. Unlock the Obsidian Tower dungeon. Clear the Obsidian Tower (50 floors). Have Grif visit the store during the evening (has to be open). Have Grif visit the next day during the evening.

Initial level: ?
Initial equipment: ?
Can equip: Claw, Clothes, Robes, Breastplate, Cloak and Boots items.

Skills
Dash: Move a short distance. Invincible while dashing. Monsters can't block.
Dark Claw: Same as Flame Rush, but black. Costs 10 SP. Character starts with this.
Dark Soul: Same as Elans Berserk skill, only lasts 15 seconds. Costs 25 SP. Character starts with this(?).
Energy Drain: Eat enemies and recover half the damage caused to the enemy. Costs 30 SP. Unlocked at level 20.

Alma
Must be in Endless Mode. Must have all the other characters unlocked. Meet her at the Store during the morning. Clear Obsidian Tower up to Floor 100. You will receive her Platinum card, which allows you to use her during all new playthroughs of the game.

Details unknown, haven't gotten her yet. Looooong list of skills though, presumably changing her skills depending on the weapon she has equipped. Can use any kind of armor.

Good info; I look forward to this FAQ's completion.

Also note the atmopshere in your store means a lot in attracting the Heroes.
Louie goes for the more plain/light look
Charme goes for the gaudy look (being a thief and all)
Caillou and Grif come in with a more of a dark look, I think Caillou is dark/gaudy while Grif is dark/plain.
I'm sure there is more for the others, but I haven't noticed. I'm sure Alma doesn't really care.

Have noticed this, but it's not a specific requirement for recruiting the characters. Just helps triggering their visit events faster.

speak for yourself lol. I spent half the game waiting for that damn mage to come in and collect his stuff. If I only knew changing my floor would have done the trick I would have had him weeks sooner.

Actually, he'll come collect them as soon as you have them. I got him before I could even change the look of my store.

just adding...

Louie seems also have high probability comes when SwordBoom, regardless the decoration , confirmed sometimes he's still come along with the Crowd of MaleTownsPeoples ,even with Gaudy Shop, when Sword Boom Anouncement + Sword on Showcase.

Charme seems also often comes when knife type weapon puttin on the show case ... since she sometimes comes when i puttin Plain Deco for luring Louie but forgot rid knife from mah showcase .

Griff only comes with dark decoration at evening/night... he hate sun badly since he was a demon, so dont waste your time waitin him on mornin/noon

Nagi managed to be lured often with Eastern Wall or Bamboo Floor... seems she missed her home , poor girl, also puttin lance type in showcase lure her tho sometimes... so far i play she only never comes with heretic decoration... which i use for luring Griff

Elan accidently figured only comes when shop atmosphere Plain/Light with Candies on the Show Case, figured why plain lure him due he's a Rival with Louie in Poor Term, also lured by light cause he was a priest in training. I still wonder why we need the candies... so far i play, he never comes if im not puttin the candies in showcase...

and yea... Arma seems just comes randomly, looks like she never care about decoration.

and if yo u need cash badly, gettin gaudy atmosphere also good for lure Alouette, that lass who self-claimed you as her rival even pay for 150% price (mean with price increased anouncement, she even willingly to pay 300%). But dont try haggin over 110% with her fairies in normal condition , often fails... i hate that fairies