You have to complete the tutorial to unlock both Cakewalk and More Pain difficulties. The hardest difficulty unlocks when you have completed the More Pain difficulty. Recommend starting on Cakewalk.
Path of the Necromancer
Complete the tutorial.
That Was Easy!
Complete the game on Cakewalk difficulty.
More Mortal Suffering!
Complete the game on More Pain! difficulty.
Evil Always Wins!
Complete the game on Good Always Wins difficulty.
Most you will get over time. You can always check progress by pressing "P" in-game. The Humonarium is a compendium of every enemy you've killed. Just keep killing to unlock all.
Unlock the Lich.
Slay 75 enemies.
Soul of Lies
Unlock the Lost Soul.
Debuff enemies 333 times.
Unlock the Head Hunter.
Defeat any three bosses.
Shadow Between Worlds
Unlock the Shade.
Trigger Insanity in 100 enemies.
Unlock the Vampire.
Turn five enemies into vampires.
Unlock the Bone Golem.
Gather 120 bones.
Unlock the Fallen Dhampir.
Lure five Dhampirs over to your side.
Unlock the Blood Phantasm.
Use 12000 Wrath on ultimate abilities.
Unlock the Mummy.
Get a Minion to level five.
Unlock the Ghoul.
Deal 5000 damage to your own minions.
Unlock all enemies in the Humonarium.
Again, you'll get this over time by completing the adventure map. If you are going after the difficulty achievements, you'll most likely die quite a bit and continue killing enemies to reach these easily.
Kill 100 enemies in the Mines.
Kill 250 enemies in the Mines.
Kill 1000 enemies in the Mines.
Mountain King I
Kill 100 enemies in the Dwarven Tunnels.
Mountain King II
Kill 250 enemies in the Dwarven Tunnels.
Mountain King III
Kill 1000 enemies in the Dwarven Tunnels.
Bane of Mercenaries I
Kill 100 enemies in the Mercenary Barracks.
Bane of Mercenaries II
Kill 250 enemies in the Mercenary Barracks.
Bane of Mercenaries III
Kill 1000 enemies in the Mercenary Barracks.
Self-explanatory. If you want to get this done easily, carry on with your tutorial save and kill all of your minions and be unable to create anymore. Killing your minions will also neat you some progress towards the other achievements as well as First Sacrifice.
I'll Be Back!
Suffer your first defeat!
Lose your first battle.
Cannon Fodder I
Lose 50 minions in battle.
Cannon Fodder II
Lose 100 minions in battle.
Cannon Fodder III
Lose 250 minions in battle.
Essentially, you'll want to upgrade all of the buildings in your Graveyard. You can see all the resources and minions you'll need by pressing "C" and clicking on one of the buildings.
Means of Recovery
Finish upgrading the Mortuary.
Finish upgrading the Arena.
Monument to Greatness
Finish upgrading the Iratus Statue.
Finish upgrading the Excavation.
Finish upgrading the Abode of Wrath.
Finish upgrading the Dead Lake.
Finish upgrading the Obelisk.
Finish upgrading the Library.
Have a minion that is composed of parts of Uncommon quality or better.
Have a minion that is composed of parts of Rare quality or better.
Have a minion that is composed of parts of Legendary quality.
Iratus and Talent Achievements
Recommended to play on a lower difficulty so your chances of losing entirely are drastically reduced.
Level up Iratus.
Have Iratus equipped with a Weapon, Armor, Ring, Amulet and Consumable.
Master of Alchemy
Learn all Alchemy talents in one playthrough.
Master of Magic
Learn all Magic talents in one playthrough.
Master of Rage
Learn all Ire talents in one playthrough.
Master of Destruction
Learn all Destruction talents in one playthrough.
You can access your alchemy table by pressing "A" in-game. There will be components on the table that you can click that allow you to transmute, distill, create, and refill mana. Simply do these things and you'll get the achievements.
Use the Alchemy table to transmute one type of part into another.
Use the Alchemy table to create an item.
Use the Alchemy table to create a part of uncommon quality or better.
Use the Alchemy table to refill your Mana.
Keeper of Keys
Defeat the Keymaster.
Defeat the Inventor.
Lord of Fire
Defeat the Pyromancer.
Defeat the Mercenary Captain.
Defeat the Treasurer.
Bane of Monsters
Defeat the Monster Slayer.