You can create a hardcore character after you reach level 10 with any regular character. When a hardcore character dies, it dies for good (permadeath), so some of the following achievements are extremely difficult to get.
Each achievement listed here is worth 10 points.
Level 10 (Hardcore)
Reach level 10 in Hardcore mode.
Achievement Reward: Banner Shape (Immortal)
Level 20 (Hardcore)
Reach level 20 in Hardcore mode.
Achievement Reward: Banner Pattern (Left Stripe)
Level 30 (Hardcore)
Reach level 30 in Hardcore mode.
Achievement Reward: Banner Accent (Imperial Crown)
Level 40 (Hardcore)
Reach level 40 in Hardcore mode.
Achievement Reward: Banner Sigil (Dark Rider)
Level 50 (Hardcore)
Reach level 50 in Hardcore mode.
Level 60 (Hardcore)
Reach level 60 in Hardcore mode.
Trophy Hunting
Achievement Rewards: Banner Sigil (Hell Steed)
Kill each type of rare enemy in Hardcore mode:
Morlu Incinerator Morlu Legionnaire Armaddon Subjugator
Mallet Lord Enslaved Nightmare Darksky Fire Demon Colossal
Golgor Succubus Soul Ripper Skull Cleaver Skull Sword
Skeletal Marauder Stygian Crawler Wintersbane Huntress Blood
Clan Occultist Blood Clan Warrior Fallen Prophet Fallen
Hellhound Fallen Soldier Fallen Grunt Fallen Slavelord Demon
Trooper Demonic Hellflyer Herald of Pestilence Armored
Destroyer Demonic Tremor Hulking Phasebeast Frenzied Hellion
Brood Hatchling Webspitter Spider Serpent Magus Noxious
Guardian Charged Construct Smoldering Construct Skeletal
Ranger Spine Hewer Bone Warrior Skeletal Raider Skeletal
Sentry Reaper Ancient Walker Sand Behemoth Sand Dweller
Dust Imp
We are the Champions
Achievement Reward: Banner Sigil (Deceiver)
Kill each type of champion in Hardcore Mode:
Morlu Incinerator Morlu Legionnaire Subjugator Mallet Lord
Enslaved Nightmaare Darksky Fire Demon Plague Swarm Succubus
Soul Ripper Dark Skeletal Bowman Skull Cleaver Skeletal
Marauder Icy Quillback Stygian Crawler Wintersbane Huntress
Blood Clan Occultist Blood Clan Sorcerer Blood Clan Impaler
Blood Clan Mauler Blood Clan Warrior Fallen Hellhound Fallen
Mongrel Fallen Soldier Fallen Grunt Fallen Slavelord Fallen
Overlord Demon Trooper Demonic Hellflyer Herald of Pestilence
Armored Destroyer Demonic Tremor Hulking Phasebeast Vile
Swarm Brood Hatchling Webspitter Spider Writhing Deceiver
Noxious Guardian Shock Guardian Chilling Construct Skeletal
Ranger Spine Hewer Bone Warrior Skeletal Raider Reaper
Sand Wasp Sand Behemoth Rock Giant Lacuni Slasher Vicious
Ghoul Grim Wraith Deathly Haunt Savage Flyer Accursed
Fallen Conjurer Fallen Cur Fallen Peon Fallen Fallen
Overseer Ravenous Dead Hungry Corpse Disentombed Hulk Unholy
Thrall Dark Hellion Dark Berserker Skeletal Archer Returned
Executioner Scavenger Quill Demon Savage Fiend Carrion
Bat Imp Horned Charger
Act I
Third Time's the Charm
Kill the Skeleton King in Hardcore mode
Skeleton King is found in the Royal Crypts under the Cathedral.
Dead Meat (Hardcore)
Achievement Rewards: Banner Accent (Chain Hooks)
Kill the Butcher on Normal Difficulty in Hardcore mode.
You receive more achievements for killing the Butcher on Hardcore Nightmare, Hardcore Hell, and Hardcore Inferno.
Butcher is in Chamber of Suffering, end of the first act.
Act II
The Witch is Dead
Kill Maghda in Hardcore mode
Kulle Story, Bro
Kill Zeltun Kulle in Hardcore mode.
No More Lies (Hardcore)
Achievement Rewards: Banner Accent (Crossed Longswords)
Kill Belial on Normal Difficulty in Hardcore mode
Additional achievements for killing Belial on harder difficulties: Hardcore Nightmare, Hardcore Hell, Hardcore Inferno
Act III
Insatiable Appetites
Kill Ghom in Hardcore mode
Aggravated Assault
Kill the Siegebreaker Assault Beast in Hardcore mode
Lust, Caution
Kill Cyadea in Hardcore mode
He Thought He Was So Clever (Hardcore)
Achievement Reward: Banner Accent (Crossed Halberds)
Kill Azmodan on Normal Difficulty in Hardcore mode
Additional achievements for killing Azmodan on Nightmare, Hell and Infernal.
Act IV
Getting Your Hope Up
Kill Rakanoth in Hardcore mode
Reign of Terror (Hardcore)
Achievement Reward: Banner Accent (Dragon)
Kill Diablo on Normal Difficulty in Hardcore mode
Also achievements for defeating Diablo on Hardcore Nightmare, Hardcore Hell, and Hardcore Inferno difficulties.