Starting Fund

Die for the first time.

Slayer I-III

Eliminate 500/2500/15000 enemies across all your runs.

Unstoppable Rampage

Deal 300000 points of melee damage in a single run.

Overkill damage does count as damage dealt. I recommend buying Great Axe at the Armory, since its spin attack can deal a lot of damage and clear in a very short amount of time.
Stacking enemy amount on higher difficulties is also a good choice.

The Bolt’s Mastery

Deal a total of 500000 points of damage with ranged weapons.

Voodoo and Hoodoo

Inflict Status Effects 1000 times in a single run.

Status effects are inflicted by elemental weapons, which are usually marked as Enchanted, mainly wands and staffs, and trigger on hit. There are 3 elements in the game and one status effect for each: Lightning (Shocks and stops the enemy for a short time), Ice (Slows down enemies) and Fire (Burning deals damage over time). Doing a run with Mage and picking wand for each element should make this achievement easy enough.
Also, enemies don’t seem to have any build ups or resistances, so you can literally stunlock enemies with shock if you have high attack speed.

Things I didn’t even try

Kill 5000 enemies with Status Effects.

I think Burning is the only Status Effect that actually deals damage, so investing into Fire weapons should be a priority.

Royal Privilege

Use Lifesteal 1000 times.

Exorcism I/Dread

Eliminate 3/20 Elite enemies.

Elites are special enemies that are tinted red. One Elite enemy always spawns on floor 10 with more showing up on later difficulties. You can spawn Elites on the upcoming floor by using or carrying certain items.

Demonslayer

Defeat 1 Succubus Elite

Succubus is one of Elite enemies. There’re 2 versions of the Succubus, one with a charge attack and one with a delayed ranged attack. Any should do.

More Powder!/Barrel Basher

Break 60/100 Barrels.

Divine Protection

Clear a Floor without taking damage 50 times.

Should be easy to do passively along the runs, especially on the early Floors.

Autoplay

Clear a Floor without moving.

Just don’t touch controls. Taking damage during this does not count as moving.

Glacial Advanced

Clear 25 Floors without dashing.

Most of the time you don’t even need to dash to avoid damage, unless you are tanking Movement Speed. With a strong build you may not even need to move at all.

Cupid

Kill 5 enemies with one ranged weapon shot.

Some of the ranged weapons, like Crossvolver or Magic Staff, pierce enemies by default, so you don’t need to play a certain class or invest much into getting it.

Ironskin

Receive 1000 damage in 1 run.

I’ll be back

Heal 100 HP on the same Floor.

Basically any build with lifesteal and high attack speed can do that.
One good strategy through recovery is by using Priestesses with Cross Wands, which give a stacking bonus to adjacent recovery items. Her Commandment of Swords can also turn armor into recovery items, adding to the bonus.

Much to Learn

Clear 5 floors without leveling up.

One way it can be done is to sell your weapons and focus purely on survival on early Floors. They are short, you don’t take that much damage, and there shouldn’t be too many enemies to overwhelm you.
Another is to tank your Exp Gain and Damage on later Floors and focus on weapons with strong Knockback or Shock to keep enemies away.

Eternal Student

Level up 5 times in one floor.

It is best done on lower floors as you get much more XP down there than later ones. Pick a fast clear weapon, like Throwing Daggers or Automatic Crossbow, stack up on scrolls and maybe try to get as close to a level up by the end of the round. One thing to note is that, apparently, Level Up Emblems found in barrels DO NOT count as a level up.

Credit to Joresh

Pain is Power/Recoverless/Holier

Win a run with 0 or less Armor/Recovery/Lifesteal.

Only the 20th Floor counts, so you can do the run as normal and then tank the stat before the boss. It is unnecessary though, as investing into one of them is enough to survive and you’ll most likely have at least one of the others tanked anyway.

Glacial

Reach -50 Speed.

Easy to do on Juggernaut and even more on Golem.

Master of the Arcane

Reach 50 Magic Power.

Just play Mage.

Bloated

Reach 25 Vitality and 25 Magic Power.

Eagle Eyes

Reach 50 Range.

Ghosted

Reach 60 Dodge.

Healthy Diet

Reach 60 Vitality.

Parasitic

Have 60% Lifesteal

Unstopable

Reach 60 armor

Knight Speed

Reach 50 Move Speed

It won’t run out

Reach 50 Luck.

Frenzied Flourish

Reach +85% Attack Speed.

Deadly Dance

Reach 65% Crit Chance.

Bulging Backpack

Have inventory size of 80.

Inventory size is tied to your level, each level granting 2 spots. Buying items that increase Exp gain and amount of enemies is the most immediate way to gain levels faster. With Titanite you can purchase levels in “Extra inventory space” stat at the Statue in the hub, which gives you one more slot on certain milestone levels. Finally, you can sometimes find a level up emblem by breaking barrels.

Payday

Have 30 Items in your inventory.

Bootstrap came off

Enter a Floor with less than 10 gold 10 times.

The very first Floor still counts, so all you need to do is keep starting gold at 0 until you get this Achievement.

Generational Wealth

Have 3000 unspent gold.

The Grand Arsenal

Equip 10 different weapons.

Concerto of Arms

Equip 12 weapons of the same kind at once.

Equipping 12 identical weapons would need a lot of space, so investing into Exp Gain and focusing on smaller arms like Daggers is recommended. Doing it with Throwing Daggers or Ebony Daggers on Rogue should be easy.

Bountiful Harvest

Have at least 6 Crimson Scythes.

Buy and keep 6 big red scythes, small ones don’t count. Do it with Fieldkeeper as she gets one for free and has bonuses for Polearms.

Edgy

Win a run with only Daggers.

This means that by the end of the run all of your weapons need to have a “Dagger” tag on them. Armor and movement items do not count, but shields do. You don’t need to do a whole run with Daggers, even though it is quite easy to do.

Rod of Lordly Might

Win a run with only staves.

“Staff” is not a set tag, so the only thing you can go for are weapons with “Staff” in the title. I don’t think that Wands count.

Less is More

Win a run with 6 or less weapons.

As stated before, you only need to have 6 weapons when you beat the boss. Shields do count as weapons.

Two More

Have 3 Tier 4 Flails.

To reliably get this you first need to buy Arcane Anvil at the Armory, which will allow you to upgrade weapons on your runs. A number of classes and aspects can get Flail as their starting weapon, and you can increase the rarity of your starting weapon to Tier 3 at the Statue with Titanite. Luck stat also increases the likelihood of higher Tier items appearing in the shop.
There is also a King unique item that makes all adjacent weapons a Tier higher, but I don’t know if you can trick the game into actually thinking them to be higher Tier.

Spoils of Conquest

Win a run with only tier 4 weapons.

Invest into Luck and Resource Gain to get a higher chance for Tier 4 weapons and money to re-roll or buy Anvils. Shields do count as weapons and cannot be upgraded with Anvils.

Power Manifest

Reach 50 Vitality and 50 Magic Power as the Vile.

Sword and Sorcery

Reach 25 Magic Power and 25 Melee Power as the Eldritch Knight.

Fire and Arms

Reach 25 Projectile Power and 25 Melee Power as the Ranger.

Sorcery and Sword

Have 5 Melee and 5 Enchanted weapons as the SpellBlade.

Enchanted Melee weapons like Prismblade and Fire Mace count towards both, so you can just focus on those.

Loyalty

Do not buy weapons for 5 Floors as the Singularist.

If you have a good weapon from the start you won’t need a new one at all. Using the Statue, you can increase the Tier of your starting weapon up to 3, and keep it until the end.
Note that shields do count as weapons.

Haul

Have 20 items as the Thief.

More

Overheal 200 times as the Soul Drinker.

When you heal on full health as Soul Drinker, you get a stacking bonus that is partially reset at the start of the new Floor. You just need to avoid damage and let it build up.

Late Night Crunch

Level up 20 times in 1 run as the Adept.

Prepared

Have an inventory of 80 as the Scholar.

Just a repeat of Bulging Backpack, but even easier.

Stand Firm

Kill 1000 enemies while not moving as the Sentinel.

Fairly easy to do with Halberds, can get Autoplay while you’re at it.

Fresh Meat

Reach 100 Vitality as the Butcher.

Can’t be Bargained

Reach 100 Armor as the Golem.

You might die of boredom on this one.

Rot

Have the first Boss be inflicted with Fire, Shock and Slow at the same time.

Wands are the most reliable for inflicting status, so pick up 3 or more wands of 3 elements and attack the boss with them. Make sure you don’t kill him too fast.

Kingly Luxury

Reach 50 Resource Gain as the King.

Riches

Spend 3000 fold in 1 Floor as the Merchant.

Just save up some money and spend them in one go. Re-rolls do count as spending.

Rags

Buy 10 Items in 1 Floor as the Beggar.

Same as the previous one, but you pay with your life.

I’m a healer, but...

Kill 1000 enemies with Blessed Mace as the Priestess

Immaculate

Take no damage for 5 consecutive Floors as the Inquisitor.

Easy to do with Cross Wand. Fully blocking damage with a shield does not count as getting hit, so you can pick up a couple just to be safe from stray projectiles. Might also be bugged, since I got it after 7 Floors.

Harvest Season

Reach 100 Range as the Fieldkeeper.

All Foes Must Die

Kill 10 Elites in one run as the Assassin.

You can spawn Elites using the items below:

• Dragonblood Elixir - Increases one of your stats by 5 and spawns one Elite on the next floor.
  
• Dust of Enlarging - Increases the size of your inventory by 3 and spawns 2 Elites on the next floor.
  
• Bloody Teeth - spawns 1 Elite every Floor as long as it remains in the inventory. Killing an Elite gives you a stacking bonus of +5% Damage and +2 Armor, even after Bloody Teeth are removed from your inventory.
  
• Malevolent Observer - Increases your Range by 40, but spawns 1 Elite every Floor.

Note that these items stack with each other and on repeated uses, as well as with Elites guaranteed to spawn due to Difficulty.

Moon in the Pool

Dodge 100 times as the Scout.

Refers to the Dodge stat, not the Dash ability. Make sure you have good healing in case RNG doesn’t work.

Divine Intervention

Block 1000 damage with a shield as the Crusader.

This refers to the actual shields and not Crusader’s aura. Buy a bunch of shields and walk into enemies.

Stick ‘em with the pointy end

Win a run as the Blade Master with only Blades

Requires you to finish a run with weapons that have a “Blade” tag. Note that Rapier and Great Sword from Blademaster’s starting weapons do not have this tag. I did this achievement with Short Swords only to be safe.

Skewer

Reach 3 Pierces as the Sharpshooter

To get this achievement, you first need to buy Masterwork Arrowhead at the Armory. Besides Arrow Shaft (which you only unlock AFTER completing this achievement), this is the only item in the game that gives Pierce. None of the stats and special effects on weapons matter.
Second, upgrade Gold Start, Free Reroll and Selling Items stats at the Statue as much as possible.
During the run, try to save up as much money as possible, as you’ll be rerolling a lot. Masterwork Arrowhead is a Tier 3 item, so it will be rare and expensive, and there is a good chance that its place will be taken up by other loot. When I did this achievement, I’ve got all 3 of my Arrowheads right on the last 2-3 Floors, but you might not get that lucky.

Master at Arm

Max out 2 weapon sets as the Weapon Master.

Each weapon has tags that indicate what set they belong to. Having multiple weapons with the same tag will give a bonus from that set. Each set maxes out at 9. Just buy the same weapon with 2 set tags 9 times until you get it.
This is also one of those achievements that have different descriptions on Steam and in the game.

Thrill of the Fight

Clear 10 Floors with less than 50% HP left as the Berserker.

The description doesn’t state this (or it is a bug), but you have to finish 10 floors on 50% HP left in a SINGLE run. Possibly consecutive floors too, which I’m not sure about.
I would not recommend doing it on lower difficulty, since there is a chance monsters won’t beat you up fast enough.
Do not take Wolf’s Fang or any other healing items. Refund your levels in “Health Gains per Floor” stat at the statue. “Pickup Range” too, so you don’t accidentally pick up a healing item or invulnerability.
Walk into enemies, explosions or stand on spikes until you get to a comfortable 30-40% HP. Projectiles are both your friends and enemies as they tend to deal far more damage than regular attacks or touch damage.
You can try to stack up on Luck and fish for a Vampire Heart in the Store.

Heaven Wills It

Defeat 5 floors in 1 run with no weapons as the Crusader.

Shields do not count as weapons for this achievement.

Loaded Dice

Reach 100% Crit Chance as the Duelist.

Ascent

Reach Floor 16.

First Step/Ascent. Again/Last Stretch/Monarch/Fun and Game

Win a Run in difficulty 1/2/3/4/5.

Note that completing a run does not require you to kill a boss on Floor 20. If your DPS is low or you are afraid of getting too close to the boss, you only need to wait for the timer to run out.

The Weapon Master

Win a run using the Weapon Master aspect of the Knight.

Secret of Swords

Win a run with all Knight aspects.

The Ranger

Win a run using the Ranger aspect of the Hunter.

The Spellblade

Win a run using the Spellblade aspect of The Wizard.

Prometheus’ Secret

Win a run with all Wizard aspects.

The Assassin

Win a run using the Assassin aspect of The Raven.

Blade in the Dark

Win a run with all Rogue aspects.

Rogue is “The Raven”, which for some reason is the only named character.

Built to Last

Win a run with all Juggernaut aspects.

Old Friend

Win a run with all Lancer aspects.

Divine Providence

Win a run with all Priestess aspects.