Here's the full list. Some of these won't be accessible until you complete another challenge to level 5. (I can't remember all of the ones that require this. All challenges that unlock another challenge are in the same category, eg. level 5 of Splish Splash in rocket launchers unlocks Catch-a-Rocket!)
Note that the description says that you do not need to do the level-specific achievements (like finding the vault symbols on each level). You only need the general, global challenges.
All challenges only need to be completed to level 1 (again, except for the ones that require level 5, which I unfortunately don't remember). Leave a comment if you see a challenge that needs to be level 5 to unlock another one.
Shout out to The Cherished for compiling the original list.
Dragon KeepScot-Free: Kill 50 dwarves.
Rock Out With Your Rock Out: Kill 10 golems.
Knighty Knight: Kill 10 knights.
Orcs Should Perish: Kill 50 orcs.
Bone Breaker: Kill 50 skeletons.
Ew Ew Ew Ew: Kill 25 spiders.
Cheerful Green Giants: Kill 10 treants.
Magical Massacre: Kill 10 wizards.
Fus Roh Die: Kill 10 dragons.
Can't Fool Me: Kill 5 mimics.
ElementalCowering Inferno: Light 25 enemies on fire.
Acid Trip: Kill 5 enemies with corrosive damage.
Boom.: Kill 5 enemies with explosives.
Slag-Licked: Deal 5000 bonus damage to slagged enemies.
Say "Watt" Again: Deal 5000 damage with shock DoT (damage-over-time) effects.
Corroderate: Deal 5000 damage with corrosive DoT (damage-over-time) effects.
I Just Want to Set the World on Fire: Deal 5000 damage with incendiary DoT (damage-over-time) effects.
LootOpen Pandora's Boxes: Open 25 lootable crates, lockers, and other objects.
Gun Runner: Pick up or purchase 10 weapons.
The Call of Booty: Open 5 treasure chests.
Nothing Rhymes with Orange: Loot or purchase 1 legendary item.
Purple Reign: Loot or purchase 1 purple-rarity item.
I Like My Treasure Rare: Loot or purchase 3 blue-rarity items.
Another Man's Treasure: Loot or purchase 10 white items.
It's Not Easy Looting Green: Loot or purchase 5 green-rarity items.
Money and TradingWhaddaya Buyin'?: Purchase 1 item with Eridium.
For the Hoard!: Save a lot (10,000) of money.
Psst, Hey Buddy...: Trade with 1 other player.
Wholesale: Sell 10 items to vending machines.
Limited-Time Offer: Buy 1 Item of the Day.
Dolla Dolla Bills, Y'all: Collect 5000 dollars from cash drops.
Health and RecoveryI'm Back! Shocked?: Get Second Winds by killing 1 enemy with a shock DoT (damage over time).
Death, Wind, and Fire: Get Second Winds by killing 1 enemy with an incendiary DoT (damage over time).
I'll Just Help Myself: Get Second Winds by killing 5 enemies.
Green Meanie: Get Second Winds by killing enemies with a corrosive DoT (damage over time).
This Is No Time for Lazy!: Revive a co-op partner 5 times.
Badass Bingo: Get Second Winds by killing 1 badass enemy.
Heal Plz: Recover 1000 health.
GrenadesPull the Pin: Kill 10 enemies with grenades.
Singled Out: Kill 5 enemies with Singularity grenades.
EXPLOOOOOSIONS!: Kill 5 enemies with MIRV grenades.
Chemical Sprayer: Kill 5 enemies with Area-of-Effect grenades.
Woah, Black Betty: Kill 5 enemies with Bouncing Betty grenades.
Health Vampire: Kill 5 enemies with 5 Transfusion grenades.
ShieldsGame of Thorns: Kill 5 enemies with reflected damage from a Spike shield.
Amp It Up: Kill 5 enemies while buffed by an Amplify shield.
Roid Rage: Kill 5 enemies while buffed by a Maylay shield.
Super Novas: Kill 5 enemies with a Nova shield burst.
Rocket LauncherSky Rockets in Flight...: Kill 10 enemies from long range with rocket launchers.
Catch-a-Rocket!: Kill 5 enemies with direct hits from rocket launchers.
Splish Splash: Kill 5 enemies with rocket launcher splash damage. (Must kill 50 enemies to unlock another challenge.)
Gone with the Second Wind: Get 1 Second Wind with rocket launchers.
Rocket and Roll: Kill 5 enemies with rocket launchers.
Sniper RifleSurprise!: Kill 5 unaware enemies with sniper rifles.
Longshot Headshot: Get 10 critical hits with sniper rifles.
Longshot: Kill 10 enemies with sniper rifles.
No Scope, No Problem: Kill 5 enemies with sniper rifles without using the scope/ironsights.
Leaf on the Second Wind: Get 1 Second Wind with sniper rifles.
Snipe Hunting: Kill 5 enemies with critical hits using sniper rifles.
Assault RifleAggravated Assault: Kill 10 enemies with assault rifles.
This Is My Rifle...: Get 10 critical hits with assault rifles.
...This Is My Gun: Kill 5 enemies with critical hits using assault rifles.
From My Cold, Dead Hands: Get 1 Second Wind with assault rifles.
Crouching Tiger, Hidden Assault Rifle: Kill 5 enemies with assault rifles while crouched.
SMGHigh Rate of Ire: Kill 5 enemies with critical hits using SMGs.
More Like Submachine FUN: Get 1 Second Wind with SMGs.
Constructive Criticism: Get 10 critical hits with SMGs.
Hail of Bullets: Kill 10 enemies with SMGs.
ShotgunShotgun!: Kill 10 enemies with shotguns.
Faceful of Buckshot: Get 25 critical hits with shotguns.
Lock, Stock, and...: Get 1 Second Wind with shotguns.
Open Wide!: Kill 10 enemies at point-blank range with shotguns. (Must get 250 kills to unlock another challenge.)
Shotgun Surgeon: Kill 5 enemies with critical hits using shotguns.
Shotgun Sniper: Kill 5 enemies from long range with shotguns.
PistolHard Boiled: Get 1 Second Wind with pistols.
Pistolero: Kill 5 enemies with critical hits.
Quickdraw: Kill 5 enemies shortly after aiming down the sights with a pistol.
Deadeye: Get 10 critical hits with pistols.
The Killer: Kill 10 enemies with pistols.
MeleeFisticuffs!: Kill 10 enemies with melee attacks.
A Squall of Violence: Kill 10 enemies with melee attacks using bladed guns.
General CombatGun Slinger: Deal 5000 damage with Tediore reloads.
Boomerbang: Kill 5 enemies with Tediore reloads.
Critical Acclaim: Kill 10 enemies with critical hits. And rainbows.
Knee-Deep in Brass: Fire 300 rounds.
...To Pay the Bills: Kill 5 enemies while using your Action Skill.
Not Full of Monkeys: Kill 5 enemies with stationary barrels.
MiscellaneousHaters Gonna Hate: Win 1 duel.
Sidejacked: Complete 1 side mission.
Yo Dawg, I Heard You Like Challenges...: Complete 5 challenges (to level 5, or a level specific challenge)
Have fun!