Hard Difficulty - defeat a boss on normal difficulty
Lunar Difficulty - defeat a boss on hard difficulty

Characters can be unlocked by defeating specific bosses on hard mode or getting a certain amount of boss kills across all difficulties.

Sniper - Beat the King's Arsenal on Hard difficulty or above (8 kills)
SpellBlade - Beat the Scholar's Nest on Hard difficulty or above (20 kills)
Bruiser - Beat the Red Darkhouse on Hard difficulty or above (30 kills)
Defender - Beat the Churchmouse Streets on Hard difficulty or above (40 kills)
Ancient - Beat the Emerald Lakeside on Hard difficulty or above (50 kills)

Sparkblade - Have a rabbit you control deal over 500 DPS in a fight (5 kills)
Ruins - Have a rabbit you control gain 3 buffs at one time (12 kills)
Timegem - Receive no rewards from a fight because you took too long to win (16 kills)
Swiftflight - Pass through the Kingdom Outskirts in less than 2 minutes (25 kills)
Darkbite - Beat any enemy in under 15 seconds (35 kills)
Sacredflame - Deal over 2500 damage in a single hit (45 kills)
Lakeshrine - Stand still doing nothing in the Emerald Lakeside for 2 full minutes (60 kills)
Gladiator - Start a boss fight alone on 2 HP or less and win (70 kills)
Youkai - Beat any enemy with exactly 77 gold on hand (80 kills)
Haunted - Apply 4 debuffs to a single enemy at once (90 kills)

Outskirts - Pass through the Kingdom Outskirts on Lunar without taking damage. (100 kills)
Nest - Defeat Twili, the Valedictorian Crow, while you are stopped in time. (120 kills)
Arsenal - Defeat Merran, the Steeltooth Wolf, while time is slowed. (140 kills)
Lighthouse - Defeat Ranalie, the Mythril Dragon, while she is dashing through the arena. (160 kills)
Streets - Defeat Matti, the Paladin Mouse, while her guard is up. (180 boss kills)
Lakeside - Defeat Avy, the Starry Frog, without standing still for longer than 5 seconds. (200 kills)
Keep - Pass through the Pale Keep on Lunar without taking damage. (220 boss kills)
Boss (Justice) - Win a run on Hard difficulty or above without a destination chosen at the start. (240 kills)
Lakeside Vocals off - Win a run on Normal difficulty or above in less than 30 minutes. (250 kills???)
Boss (Serious) - Win a run on Normal difficulty or above without opening treasure chests. (275 kills)
Boss (Mysterious) - Win a run on Normal difficulty or above without talking to Asha, the Shopkeeper Cat. (300 kills)
Pinnacle - Win a run on Normal difficulty or above without taking damage. (325 kills)
Ending - Save Shira from the Spell, (350 kills)

I believe everything can be unlocked by killing a certain amount of bosses. All that's left now is lakeside vocals which I put a 250??

Thanks for all the information in the comments!