To unlock the Bounty Hunter trophy in Borderlands 2, you will need to complete 20 side missions.
You can tell the difference between the main missions and side missions as they'll say "Optional" next to them. Complete 20 from the list below and the trophy will unlock.
List taken from the Xbox 360 guide by Asqueerj:
*** Spoiler - click to reveal ***3:10 to Kaboom
A Real Boy: Clothes Make the Man
A Real Boy: Face Time
A Real Boy: Human
Animal Rescue: Food
Animal Rescue: Medicine
Animal Rescue: Shelter
Animal Rights
Arms Dealing
Assassinate the Assassins
Bad Hair Day
Bandit Slaughter: Round 1
Bandit Slaughter: Round 2
Bandit Slaughter: Round 3
Bandit Slaughter: Round 4
Bandit Slaughter: Round 5
Bearer of Bad News
Best Mother's Day Ever
BFFs
Breaking the Bank
Capture the Flags
Clan War: End of the Rainbow
Clan War: First Place
Clan War: Reach the Dead Drop
Clan War: Starting the War
Clan War: Trailer Trashing
Clan War: Wakey Wakey
Clan War: Zafords vs. Hodunks
Claptrap's Birthday Bash
Claptrap's Secret Stash
Cult Following: Eternal Flame
Cult Following: False Idiots
Cult Following: Lighting the Match
Cult Following: The Enkindling
Customer Service
Defend Slab Tower
Demon Hunter
Do No Harm
Doctor's Orders
Get to Know Jack
Handsome Jack Here
Hell Hath No Fury
Hidden Journals
Home Movies
Hungry Like the Skag
Hyperion Contract 873
Hyperion Slaughter: Round 1
Hyperion Slaughter: Round 2
Hyperion Slaughter: Round 3
Hyperion Slaughter: Round 4
Hyperion Slaughter: Round 5
In Memoriam
Kill Yourself
Medical Mystery
Medical Mystery: X-Com-municate
Mighty Morphin
Mine All Mine
Minecart Mischief
Monster Mash: Part 1
Monster Mash: Part 2
Monster Mash: Part 3
Neither Rain nor Sleet nor Skags
No Hard Feelings
No Vacancy
Note for Self-Person
Out of Body Experience
Perfectly Peaceful
Poetic License
Positive Self Image
Rakkaholics Anonymous
Rock, Paper, Genocide: Corrosive Weapons!
Rock, Paper, Genocide: Fire Weapons!
Rock, Paper, Genocide: Shock Weapons!
Rock, Paper, Genocide: Slag Weapons!
Rocko's Modern Strife
Safe and Sound
Shielded Favors
Shoot This Guy in the Face
Showdown
Slap-Happy
Splinter Group
Stalker of Stalkers
Statusque
Swallowed Whole
Symbiosis
The Bane
The Chosen One
The Cold Shoulder
The Good, the Bad, and the Mordecai
The Great Escape
The Ice Man Cometh
The Lost Treasure
The Name Game
The Overlooked: Medicine Man
The Overlooked: Shields Up
The Overlooked: This is Only a Test
The Pretty Good Train Robbery
This Just In
This Town Ain't Big Enough
To Grandmother's House We Go
Too Close for Missiles
Torture Chairs
Uncle Teddy
Won't Get Fooled Again
Written by the Victor
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