Successfully complete one of the following tasks to get a trophy:

Platinum (Platinum): Collect all other 55 Trophies for this Trophy.
Relic Finder (Bronze): Find the strange relic.
Survivor (Bronze): Defeat 75 Enemies in a row without dying.
100 Headshots (Bronze): Defeat 100 enemies with headshots.
250 Headshots (Silver): Defeat 250 enemies with headshots.
Run-and-Gunner (Bronze): Defeat 20 enemies by shooting from the hip (without aiming).
Hangman (Bronze): Defeat 20 enemies with gunfire by aiming while hanging.
50 Kills: 92FS - 9mm (Bronze): Defeat 50 enemies with the 92FS - 9mm.
50 Kills: Micro - 9mm (Bronze): Defeat 50 enemies with the Micro - 9mm.
30 Kills: Wes - 44 (Bronze): Defeat 30 enemies with the Wes - 44.
30 Kills: Desert - 5 (Bronze): Defeat 30 enemies with the Desert - 5.
70 Kills: FAL (Bronze): Defeat 70 enemies with the FAL.
70 Kills: M4 (Bronze): Defeat 70 enemies with the M4.
30 Kills: Moss - 12 (Bronze): Defeat 30 enemies with the Moss - 12.
70 Kills: SAS - 12 (Bronze): Defeat 70 enemies with the SAS - 12.
50 Kills: M79 (Bronze): Defeat 50 enemies with the M79.
50 Kills: GP32 - BND (Bronze): Defeat 50 enemies with the GP32 - BND.
50 Kills: Dragon Sniper (Bronze): Defeat 50 enemies with the Dragon Sniper.
30 Kills: RPG - 7 (Bronze): Defeat 30 enemies with the RPG - 7.
200 Kills: GAU - 19 (Bronze): Defeat 200 enemies with the GAU - 19.
30 Kills: Mk - NDI (Bronze): Defeat 30 enemies with the Mk - NDI.
Bare-knuckle Slugger (Silver): Defeat 50 enemies with hand-to-hand combat.
Master Ninja (Silver): Defeat 50 enemies with stealth attacks.
Steel Fist Master (Bronze): Defeat 20 enemies with a single punch, after softening them up with gunfire.
Steel Fist Expert (Silver): Defeat 10 enemies in a row with a single punch, after softening them up with gunfire.
Charted! - Easy (Bronze): Finish the game in Easy Mode.
Charted! - Normal (Silver): Finish the game in Normal Mode.
Charted! - Hard (Gold): Finish the game in Hard Mode.
Charted! - Crushing (Gold): Finish the game in Crushing Mode.
Secrets of the Kuna (Bronze): Complete the Mystery.
Odessa Mining Company (Bronze): Complete the Mystery.
The Conquistadors (Bronze): Complete the Mystery.
Proof of Life (Bronze): Complete the Mystery.
Poisoned Powder (Bronze): Complete the Mystery.
The Revolution (Bronze): Complete the Mystery.
Trail of Vincent Perez (Bronze): Complete the Mystery.
The Sete Cidades (Bronze): Complete the Mystery.
The Friar's Pilgrimage (Bronze): Complete the Mystery.
The Lost Civilization (Bronze): Complete the Mystery.
The Circle of Heaven (Bronze): Complete the Mystery.
The Ring of Earth (Bronze): Complete the Mystery.
Treasure: Turquoise Glyphs (Bronze): Complete the treasure set.
Treasure: Jade Carvings (Bronze): Complete the treasure set.
Treasure: The Menagerie (Bronze): Complete the treasure set.
Treasure: Minor Deities (Bronze): Complete the treasure set.
Bounty: Arcana (Bronze): Complete the bounty set.
Bounty: Pieces of Silver (Bronze): Complete the bounty set.
Bounty: Cádiz (Bronze): Complete the bounty set.
Bounty: Gemstones (Bronze): Complete the bounty set.
Puzzle Master (Bronze): Complete all puzzles.
Rub One Out (Bronze): Complete all rubbings.
Paparazzo (Bronze): Complete all photographs.
Touch My Rear (Bronze): Use the rear touch pad to climb a rope or chain.
The Collector (Bronze): Collect first treasure or mystery.

Additionally, there is one secret trophy:

Ruffle My Feathers (Bronze): Force José Parrot to squawk out 8 unique quips.