This is a complete list of the challenges required for this achievement. I assume most of these are self explanatory. If it says "kill an enemy with X" then the actual kill shot must be that (i.e. elements). If you're farming kills for any of these, simply find the weapons and head to a low-level area. If you passed up the arenas and are going back, that's a good place to stack kills.
*** Beware as some aren't unlocked until other challenges are completed.
Enemies:
- Little Person, Big Pain: Kill 10 Midgets
- Load and Lock: Kill 20 Loaders
- Bully the Bullies: Kill 25 Bullymongs
- Cruising for a Bruising: Kill 10 Bruisers
- Marauder? I Hardly Know 'Er: Kill 20 Marauders
- You Dirty Rat: Kill 10 Rats
- Mama's Boys: Kill 50 Psychos
- You (No)Mad, Bro?: Kill 10 Nomads
- Short-Chained: Shoot a chain to release midgets from shield
- Hurly Burly: Shoot 10 bullymong-tossed projectiles out of midair
- Just a Moment of Your Time...: Kill 10 surveyors
- WHY SO MICH HIRT?!: Kill 10 goliaths
- Paigineering: Kill 10 Hyperion personnel
- Constructor Destructor: Kill 5 constructors
- Another Bug Hunt: Kill 10 varkids (Unlocked with Tundra Express)*
- Tentacle Obsession: Kill 10 threshers (Unlocked with Highlands - Outwash)*
- Pest Control: Kill 10 spiderants (Unlocked with Frostburn Canyon)*
- Crystals Are a Girl's Best Friend: Kill 10 crystalisks (Unlocked with The Fridge)*
- Pod Pew Pew: Kill 10 Varkid pods before they hatch (Unlocked with Tundra Express)*
- Die in the Friendly Skies: Kill 10 buzzards (Unlocked with The Dust)*
- You're One Ugly Mother...: Kill 10 stalkers (Unlocked with The Highlands - Outwash)
- Skags to Riches: Kill 10 skags (Unlocked with Three Horns - Divide)*
Elemental:
- Slag-Licked: Deal 5000 bonus damage to Slagged enemies
- Boom.: Kill 20 enemies with explosive damage
- Cowering Inferno: Ignite 25 enemies
- Say 'Watt' Again: Deal 5000 electrocute damage
- Corroderate: Deal 2500 corrosion damage
- I Just Want to Set the World on Fire: Deal 5000 burn damage
- Acid Trip: Kill 20 enemies with corrode damage
Loot:
- Open Pandora's Boxes: Open 50 lootable chests, lockers, and other objects
- The Call of Booty: Open 5 treasure chests
- Nothing Rymes with Orange: Loot or purchase 1 orange item
- Purple Reign: Loot or purchase 2 purple items
- I Like My Treasure Rare: Loot or purchase 5 blue items
- It's Not Easy Looting Green: Loot or purchase 20 green items
- Another Man's Treasure: Loot or purchase 50 white items
- Gun Runner: Pick up or purchase 10 weapons
Money & Trading:
- Psst, Hey Buddy...: Trade with 1 other player
- Whaddaya Buyin'?: Purchase 2 items with Eridium
- Limited-Time Offer: Buy 1 "Item of the Day"
- Wholesale: Sell 10 items to vending machines
- For the Hoard!: Save $10,000
- Dolla Dolla Bills, Y'all: Pickup $5,000 from cash drops
Vehicle:
- Passive Aggressive: Kill 1 enemy while riding as a passenger
- Turret Syndrome: Kill 10 enemies with a turret or vehicle mounted weapon
- Blue Sparks: Kill 5 enemies while power sliding (Hold RB)
- Hit-and-Fun: Kill 5 enemies by ramming them with a vehicle
- ...One Van Leaves - Kill 5 vehicles while in a vehicle
Health & Recovery:
- I'll Just Help Myself: Get 5 "Second Winds" by killing an enemy
- This Is No Time for Lazy!: Revive a co-op partner 5 times
- Badass Bingo: Get 1 "Second Wind" by killing a badass enemy
- Heal Plz: Heal 1000 points worth of damage
- Death, Wind, and Fire: Get 1 "Second Wind" by killing an enemy with burn damage over time (Unlocked after Cowering Inferno level 1)*
- Green Meanie: Get 1 "Second Wind" by killing an enemy with corrosive damage over time (Unlocked after completing Acid Trip level 1)*
- I'm Back! Shocked?: Get 1 "Second Wind" by killing an enemy by electrocution damage over time (Unlocked after Say "Watt" Again level 1)*
Grenades:
- Woah, Black Betty: Kill 10 enemies with Bouncing Betty grenades
- Health Vampire: Kill 10 enemies with Transfusion grenades
- Pull the Pin: Kill 10 enemies with grenades
- Chemical Sprayer: Kill 10 enemies with "Area of Effect" grenades
- EXPLOOOOOSIONS!: Kill 10 enemies with Mirv grenades
- Singled Out: Kill 10 enemies with Singularity grenades
Shields:
- Ammo Eater: Absorb 20 ammo with an Absorption shield
- Roid Rage: Kill 5 enemies while buffed by a "Maylay" shield
- Game of Thorns: Kill 5 enemies with reflected damage from a Spike shield
- Amp It Up: Kill 5 enemies while buffed by an Amplify shield
- Super Novas: Kill 5 enemies with a Nova burst shield
Rocket Launcher:
- Rocket and Roll: Kill 10 enemies with rocket launchers
- Gone with the Second Wind: Get 2 "Second Winds" with rocket launchers
- Sky Rockets in Flight...: Kill 25 enemies from long range with rocket launchers (Unlocked after Gone With The Second Wind Level 1)*
- Shield Basher: Kill 5 shielded enemies with 1 rocket each (Unlocked after Splish Splash level 1)
- Splish Splash - Kill 200 enemies with rocket launcher splash damage (Unlocked after Rocket And Roll level 1)*
- Catech-a-Rocket: Kill 5 enemies with direct rocket hits (Unlocked after splish Splash level 5)*
Sniper Rifle:
- Longshot: Kill 20 enemies with sniper rifles
- Longshot Headshot: Get 25 critical hits with sniper rifles
- Leaf on the Second Wind: Get 2 "Second Winds" using sniper rifles
- Eviscerated: Kill 5 shielded enemies with 1 sniper rifle shot (Unlocked after Longshot level 1)*
- No Scope, No Problem: Kill 5 enemies without using zoom (Unlocked after Longshot level 1)*
- Snipe Hunting - Kill 10 enemies with critical hits using sniper rifles Unlocked after Longshot Headshot level 1?)*
- Surprise! - Kill 5 unaware enemies with sniper rifles (Unlocked after No Scope, No Problem level 1)*
Assault Rifle:
- Aggravated Assault: Kill 25 enemies with assault rifles
- From My Cold, Dead Hands: Get 5 "Second Winds" using assault rifles
- This is My Rifle...: Get 25 critical hits with assault rifles
- ...This is My Gun: Kill 10 enemies with critical hits using assault rifles (Unlocked after This Is My Rifle... level 1)*
- Crouching Tiger, Hidden Assault Rifle: Kill 25 enemies with assault rifles while crouched (Unlocked after Aggravated Assault level 1)*
SMG:
- Hail of Bullets: Kill 25 enemies with SMGs
- Constructive Criticism: Get 25 critical hits with SMGs
- High Rate of Ire: Kill 10 enemies with critical hits using SMGs
- More Like Submachine FUN: Get 2 "Second Winds" with SMGs (Unlocked after Constructive Criticism level 1)*
Shotgun:
- Faceful of Buckshot: Get 50 critical hits with shotguns
- Shotgun!: Kill 25 enemies with shotguns
- Lock, Stock, and ...: Get 2 "Second Winds" with shotguns
- Shotgun Surgeon: Kill 10 enemies with critical hits using shotguns
- Open Wide!: Kill 750 enemies from point-blank range with shotguns
- Shotgun Sniper: Kill 10 enemies from long range with shotguns (Unlocked after open Wide! level 5)*
Pistol:
- Hard Boiled: Get 2 "Second Winds" with pistols
- The Killer: Kill 25 enemies with pistols
- Deadeye: Get 25 critical hits with pistols
- Quickdraw: Kill 10 enemies shortly after entering ironsights (Unlocked after The Killer level 1)*
- Pistolero: Kill 10 enemies with critical hits using pistols (Unlocked after Deadeye level 1)*
Melee:
- Fisticuffs!: Kill 25 enemies with melee attacks
- A Squall of Violence: Kill 20 enemies with bladed guns (Unlocked after Fisticuffs! level 1)*
General Combat:
- Knee-Deep in Brass: Fire 1000 rounds
- Afternoon Delight: Kill 50 enemies during the day
- ...To Pay the Bills: Kill 20 enemies while using your action skills
- Critical Acclaim: Kill 20 enemies with critical hits
- ...I got to Boogie: Kill 10 enemies at night
- Boomerbang: Kill 5 enemies with Tediore reloads
- Gun Slinger: Deal 5,000 damage with Tediore reloads
- Not Full of Monkeys: Kill 10 enemies with stationary barrels
Miscellaneous:
- JEEEEENKINSSSSSS!!!: Kill Jimmy Jenkins once (See RedemtionOWNAGE's solution for more information)
- Sidejacked: Complete 5 side missions
- Compl33tionist: Complete 10 optional mission objectives
- Haters Gonna Hate: Win 1 duel
- Yo Dawg I Herd You Like Challenges: Complete 5 challenges
Credit goes to FrustratedNerd on trueachievements for posting this list