Decided to compile a list of games that require you to pay for extra content beyond the initial game to unlock one or more achievements.
Under each game you'll notice I listed out any special editions/bundles/packs that the DLC is included in plus other various notes like if the content is still buyable or if different versions also have the DLC.

List is still a WIP so I probably missed a ton of stuff.
If you see any games I don't have that need DLC please let me know.
I also haven't played any of the Steam MMO's that require subscriptions so if anyone knows of any please let me know about them. (DC Universe comes to mind, I know PlayStation requires one but not sure if it's the same for PC)
If you see any info that needs to be corrected please let me know.

Batman: Arkham City
[Game and DLC no longer directly buyabale on Steam]
-Catwoman Bundle Pack
-Robin Bundle Pack
-Nightwing Bundle Pack
-Harley Quinn's Revenge

Batman: Arkham Knight
(Batman: Arkham Knight Premium Edition)
[These DLC's + other cosmetic extras available with the Season Pass]
-Red Hood Story Pack
-Harley Quinn Story Pack
-Batgirl: A Matter of Family
-1989 Movie Batmobile Pack
-Batman Classic TV Series Batmobile Pack
-GCPD Lockdown
-2008 Tumbler Batmobile Pack
-Catwoman's Revenge
-A Flip of a Coin
-2016 Batman v Superman Batmobile Pack
-Season of Infamy
-Original Arkham Batmobile

Batman: Arkham Origins
[These DLC's + other cosmetic extras available with the Season Pass]
-Initiation
-Cold, Cold Heart

BioShock 2
-Minerva’s Den

BioShock Infinite
(Bioshock Infinite + Season Pass Bundle ROW)
[These DLC's available via Season Pass if you own base game]
-Clash in the Clouds
-Burial at Sea - Episode 1
-Burial at Sea - Episode 2

Borderlands
(Borderlands: Game of the Year)
-The Zombie Island of Dr. Ned
-Mad Moxxi's Underdome Riot
-The Secret Armory of General Knoxx
-Claptrap's New Robot Revolution

Borderlands 2
(Borderlands 2 Game of the Year)
[These DLC's available via Season Pass if you own base game]
[Alternate regional version has same DLC]
-Captain Scarlett and her Pirate's Booty
-Mr. Torgue’s Campaign of Carnage
-Sir Hammerlock's Big Game Hunt
-Tiny Tina’s Assault on Dragon Keep

CastleStorm
(Castlestorm Complete Edition)
-From Outcast to Savior
-The Warrior Queen

Dead Island
(Dead Island: Game of the Year Edition)
[DLC not buyable stand alone]
-Bloodbath Arena

Defense Grid: The Awakening
(Defense Grid: Containment Bundle)
-Containment DLC
-You Monster DLC
-Resurgence Map Pack 1
-Resurgence Map Pack 2
-Resurgence Map Pack 3
-Resurgence Map Pack 4

Duke Nukem Forever
(Duke Nukem Forever Collection)
[Alternate regional version has no DLC]
-The Doctor Who Cloned Me

The Elder Scrolls V: Skyrim
(The Elder Scrolls V: Skyrim - Legendary Edition)
-Dawnguard
-Hearthfire
-Dragonborn

Fallout 4
[These DLC's available via Season Pass if you own base game]
-Not released yet

Fallout: New Vegas
(Fallout: New Vegas Ultimate Edition)
[Alternate regional version has same DLC]
-Dead Money
-Honest Hearts
-Old World Blues
-Lonesome Road
-Gun Runners Arsenal

L.A. Noire
(L.A. Noire Complete Edition)
[These DLC's available via L.A. Noire: DLC Bundle if you own base game]
-The Naked City
-A Slip of the Tongue
-Reefer Madness
-Nicholson Electroplating

Mafia II
(Mafia II: Digital Deluxe Edition)
[Alternate regional version has same DLC]
[Game and DLC currently not buyable directly on Steam]
-Jimmy's Vendetta
-Joe's Adventures

Magic: The Gathering - Duels of the Planeswalkers 2012
(Magic 2012 Gold Game Bundle)
-Magic: The Gathering 2012 - Expansion

Magic: The Gathering - Duels of the Planeswalkers 2013
(Magic 2013 Gold Game Bundle -or- Magic 2013 Complete Bundle)
-Magic: The Gathering - Duels of the Planeswalkers 2013 Expansion

The Maw
(The Mawesome Pack)
-Brute Force
-River Redirect
-Speeder Lane

Max Payne 3
(Max Payne 3 Complete)
[These DLC's + other extras available via Max Payne 3 Rockstar Pass if you own base game]
-Local Justice Pack
-Hostage Negotiation Pack
-Painful Memories Pack
-Deathmatch Made in Heaven Pack

Metro: Last Light
(Metro: Last Light Complete Edition)
[Base game and Complete Edition no longer buyable directly on the Steam store]
[These DLC's available via Season Pass if you own base game]
[Alternate regional version has the Ranger Mode DLC] *Need confirmation if this is DLC or apart of the Asian version base game*
-Ranger Mode
-Faction Pack
-Tower Pack
-Developer Pack
-Chronicles Pack

Middle-earth: Shadow of Mordor
(Middle-earth: Shadow of Mordor Game of the Year Edition)
-Lord of the Hunt
-The Bright Lord

PC Gamer
[Game and DLC no longer directly buyabale on Steam]
-

Phantom Breaker: Battle Grounds
-Phantom Breaker: Battle Grounds - Kurisu Makise + Level 99 Pack

Resident Evil 5
(Resident Evil 5 Gold Edition)
-Resident Evil 5 - UNTOLD STORIES BUNDLE

Resident Evil 6
(Resident Evil 6 Complete)
-Onslaught Game Mode
-Survivors Game Mode
-Predator Game Mode
-Siege Game mode

Rise of the Tomb Raider
(Rise of the Tomb Raider - Digital Deluxe Edition)
[These DLC's available via Season Pass if you own base game]
-Endurance Mode
-Baba Yaga

Risen 2: Dark Waters
(Risen 2: Dark Waters Gold Edition -or- Risen Franchise Pack)
[Alternate regional version has same DLC]
-Treasure Isle
-Air Temple

Risen 3: Titan Lords
(Risen 3 - Complete Edition -or- Risen Franchise Pack)
-Fog Island
-Uprising of the Little Guys

Saints Row: The Third
(Saints Row: The Third - The Full Package -or- Saints Row Ultimate Franchise Pack)
[These DLC's available via Season Pass if you own base game]
-Genkibowl VII
-Gangstas in Space
-The Trouble With Clones

Saints Row IV
(Saints Row IV: Game of the Century Edition -or- Saints Row Ultimate Franchise Pack -or- Saints Row / Metro Double Pack)
[These DLC's available via Season Pass if you own base game]
-Enter the Dominatrix
-How the Saints Save Christmas

Serious Sam 3: BFE
(Serious Sam 3 BFE Gold -or- Serious Sam Complete Pack)
-Serious Sam 3: Jewel of the Nile

Sleeping Dogs
[Game no longer buyable directly on Steam but DLC's still available]
-Nightmare in North Point Pack
-Year of the Snake

Swords and Soldiers HD
(Swords and Soldiers + Super Saucy Sausage Fest DLC)
-Swords and Soldiers - Super Saucy Sausage Fest DLC

State of Decay
[Game no longer buyable directly on Steam but DLC's still available]
-Breakdown
-Lifeline

Tom Clancy's Ghost Recon: Future Soldier
[These DLC's available via Season Pass if you own base game]
-Artic Strike
-Raven Strike
-Khyber Strike

The Walking Dead
(Telltale Games Collection)
-The Walking Dead: 400 Days

Winter Voices
(Winter Voices Complete Pack)
-Winter Voices Episode 1: Those who have no name
-Winter Voices Episode 2: Nowhere of me
-Winter Voices Episode 3: Like a Crow on a Wire
-Winter Voices Episode 4: Amethyst Rivers
-Winter Voices Episode 5: Overflow
-Winter Voices Episode 6: Falls

The Witcher 3: Wild Hunt
(The Witcher 3: Wild Hunt Game + Expansion Pass)
[These DLC's available via Expansion Pass if you own base game]
-Hearts of Stone

Worms Revolution
(Worms Revolution Gold Edition -or- Worms Collection)
-Mars Pack

Worms Ultimate Mayhem
(Worms Ultimate Mayhem - Deluxe Edition -or- Worms Collection)
-Multi Player Pack
-Single Player Pack