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.