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

Welcome to the Jungle (Bronze): Kill 50 Deinonychus.
Woah (Bronze): Kill 1 dinosaur during an Adrenaline Burst.
Preparing for the Future (Bronze): Repair all broken entrances before unlocking the gate.
Pack Hunting (Bronze): Kill 3 Raptors with one grenade.
Who Needs a Bug Zapper? (Bronze): Kill 50 Pterosaurs in Rise Above.
Death from Below (Bronze): Kill 1 Pterosaur with a grenade.
Blast in the Past (Bronze): Kill 20 Dinosaurs with grenades.
Plumber Stomachache (Bronze): Destroy 50 exploding mushrooms.
1-Up (Bronze): Kill 1 enemy using an exploding mushroom.
Crispy Critters (Bronze): Kill 5 Scorpions with the RPG.
Scorpion King (Bronze): Kill 50 Scorpions.
Chicken Chaser (Bronze): Kill 50 Velociraptors.
Raptor Extinction (Silver): Kill 100 Deinonychus.
Not so Fast (Silver): Kill 3 Raptors during a single Adrenaline Burst.
Adrenaline Junkie (Silver): Kill 25 dinosaurs during Adrenaline Bursts.
Gunman (Silver): Kill a dinosaur using each weapon.
Firepower (Silver): Kill 50 dinosaurs using stationary weapons.
Dinomite! (Silver): Kill 50 Raptors with a single shot each.
Six Foot Turkey (Silver): Kill a Utahraptor.
Outta Nails (Silver): Complete any fortification objective without repairing.
Handyman (Silver): Complete any fortification objective without the dinosaurs getting in.
These are not the Hammer (Gold): Kill 20 dinosaurs using melee attacks.
What a Prick (Gold): Survive a Spinosaurus attack.
I'm the King (Gold): Kill a Tyrannosaurus.
Nuck Chorris Rex (Gold): Complete the game without dying.
Life Finds a Way (Gold): Complete the game.
Hard as Nails (Gold): Complete the game on the Hard difficulty.
Platinum Trophy (Platinum): Get all trophies.

Additionally, there are four secret trophies:

Die in a Fire (Unknown): Killed yourself by fire or lava.
Allergic Reaction (Unknown): Kill yourself by thorny plants.
Clever Girl (Unknown): Die in a Raptor Melee QTE.
Mari-Ouch (Unknown): Kill yourself with an exploding mushroom.