Achievements for killing the Skeleton King
The Skeleton King can be found under the Cathedral, in the Royal Crypts. It is the first real boss you will fight. Each of these achievements is worth 10 points.

The Mad King's End:
Kill the Skeleton King on Normal Difficulty (Reward: Banner Shape)
The Mad King's End (Nightmare): Kill the Skeleton King on Nightmare Difficulty (Reward: Banner Accent)
The Mad King's End (Hell): Kill the Skeleton King on Hell Difficulty (Reward: Banner Pattern)
The Mad King's End (Inferno): Kill the Skeleton King on Inferno Difficulty (Reward: Banner Sigil)

The Last Stand of the Ancients (10 Points)
Complete the Last Stand of the Ancients event

This event takes place in the Festering Woods, look for stairs leading up to a memorial stone, you will then have to defeat some enemies that attack.

The Matriarch's Bones
(10 Pts)
Complete the Matriarch's Bones event

This is an event in one of the tombs under the Cemetery of the Forsaken.

Eternal War (10)
Complete the Eternal War event

This event starts in the Festering Woods. Interact with the Nephalem Monument then kill the attacking ghouls.

Revenge of Gharbad (10)
Complete the Revenge of Gharbad event

This takes place in the Ancient Pyre near the Southern Highlands. Talk to the Ghost of Gharbad to begin this event

Thorough Investigation (10)
Explore the following areas of the Cathedral:
Cathedral Level 1
Leoric's Passage
Cathedral Garden
Cathedral Level 2
Cathedral Level 3
Cathedral Level 4
The Royal Crypts

You will be exploring all these areas as you advance in the story.

Field Work (10)
Explore the following areas of the Fields of Misery
The Weeping Hollow
Cemetery of the Forsaken
Fields of Misery
Sheltered Cottage
The Old Mill
Drowned Temple
The Festering Woods
Crypt of the Ancients
Warrior's Rest

Butchered! (10)
Achievement Rewards: Banner Shape, Accent, Pattern Sigil
Kill the Butcher on Normal Difficulty (And also achievements for killing on Nightmare, Hell and Inferno)

The Butcher is in the Chamber of Suffering

Carrion Farm (10)
Complete the Carrion Farm event

This is an event in the Forsaken Grounds

Jar of Souls (10)
Complete the Jar of Souls event

This is in one of the tombs under Cemetery of the Forsaken, find a jar atop a pedestal to start the event.

A Nice Place to Visit (10)
Explore the following ares of New Tristram
Overlook Road
New Tristram
The Slaughtered Calf Inn
Leah's Room
Cain's House
Old Tristram Road
The Old Ruins
Adria's Hut

Note: Leah's Room will unlock in a cutscene after doing the Doom in Wortham quest

In the Mountains (10)
Explore the following areas of the Highlands: Wortham, Wortham Chapel Cellar, Wortham Bluffs, Caverns of Araneae, Chamber of Queen Araneae, Highlands Crossing, Southern Highlands, Karyna's Lost Wagon, Northern Hughlands, Leoric's Hunting Grounds, Leoric's Manor Courtyard, Leoric's Manor

The main story will take you to all the above places needed for this achievement

There will Definitely be Blood (10)
Explore the following areas of the Halls of Agony
Halls of Agony Level 1
Halls of Agony Level 2
Highlands Passage
The Cursed Hold
Halls of Agony Level 3
Chamber of Suffering

No Stone Unturned (10)
Explore the following areas of the Halls of Agony
IMPORTANT: The locations of all these dungeons can be found in the following guide: Diablo III Act 1 Random Dungeons Guide
Lost Mine Level 1, Lost Mine Level 2, Scavenger's Den Level 1, Scavenger's Den Level 2, Cave of the Moon Clan Level 1, Cave of the Moon Clan Level 2, Highlands Cave, Den of the Fallen Level 1, Den of the Fallen Level 2, The Lyceum, The Cave Under the Well, Watch Tower Level 1, Watch Tower Level 2, Decaying Crypt Level 1, Decaying Crypt Level 2, House of Curios, abandoned Servant House, Musty Cellar, Dank Cellar, Damp Cellar, Dark Cellar, Mass Grave, Farmer's Cellar, Tinker's Hovel, Pillaged Home

Wanderer of Tristram (10)
Achievement Rewards: Banner Accent
Complete the Act I exploration achievements listed below:
A Nice Place to Visit
Thorough Investigation
Field Work
In the Mountains
There will Definitely be Blood
No Stone Unturned

A Quick Study (10)
Read the following people Lore Books in Act I:
Adria's Journal, Lachdanan's Scroll Part 1, Lachdanan'd Scroll Part 2, Lachdanan's Scroll Part 3, Lachdanan's Scroll Part 4, Lachdanan's Scroll Part 5, Leah's Journal Part 1, Leah's Journal Part 2, Leah's Journal Part 3, Leah's Journal Part 4. Leah's Journal Part 5, Leah's Journal Part 6, Leah's Journal Part 7, Leoric's Journal Part 1, Leoric's Journal Part 2, Leoric's Journal Part 3, Leoric's Journal Part 4, Leoric's Journal Part 5, Farmer's Journal, Scabbard of Talic the Defender, Deckard Cain's Journal Part 1, Deckard Cain's Journal Part 2, Adrenah's Jottings, Hermit's Ravings, A Prisoner's Journal, Queen Asylla's Journal, Scoundrel's Journal, Kingsport Bill, The Skeleton King, Wandering Tinker's Diary, Notes of Urik the Seer Part 1, Notes of Urik the Seer Part 2, Notes of Urik the Seer Part 3, Priest's Contemplations Part 1, Priest's Contemplations Part 2, Mira's Letter to Haedrig, Cain's Old Journal Part 1, Cain's Old Journal Part 2, Gillian's Diary Part 1, Gillian's Diary Part 2, Lost Journal, Old Diary, Despina's Journal

Taking Notes (10)
Read the following Quest Lore Books in Act I:
Report from Wortham
Command from Cultist Grand Inquisitor
Reply from Cultist Grand Inquisitor
A Page from Lazarus's Grimioire
Orders from Maghda
Villager's Journal
A Summoner's Journal

Historian of Tristram (10)
Read the following World Lore Books in Act I
The Hanging Tree, Gravedigger's Log, Tristram Fields, Khazra Heads Wanted, Researching the Khazra, The Khazra Massacre, The Highlands, Torn Letter, Torn Letter's Response, New Tristram, Traveler's Journal, Warriv's Journal, Tomb Robber's Journal, The Last Stand of the Ancients, The Drowned Temple, Warrior's Rest, Old Tristram Journal

Beastmaster of Tristram (10)
Read the following Bestiary Lore Books in Act I
Beast, Dark Berserker, Carrion Bat, Carroin Nest, Imp, Dark Cultist, Ghost, Ghoul, Grotesque, Hellion, Khazra Shaman, Quill Fiend, Scavenger, Skeleton, Spiderling, Spider, Treasure Goblin, Unburied, Dark Vessel, Withermoth, Wood Wraith, Wretched Mother, Risen Dead, Butcher, The Spider Queen

The Last of the Horadrim (10)
Achievement Rewards: Banner Shape
Complete all quests in Act I
The Fallen Star, The Legacy of Cain, A Shattered Crown, Reign of the Black King, Sword of the Stranger, The Broken Blade, The Doom in Wortham, Trailing the Coven, The Imprisoned Angel, Return to New Tristram

Use our Walkthrough to complete all the quests for this achievement.