There are 11 Trophies that can be collected.

Bronze

First Strike
Be the first to get an objective in a game.

Destroyer
Kill 100 human, cat, robot, yeti and zombie units.

Faction Specialist
Complete the human, cat, robot, yeti and zombie campaigns.

Soldier
Play 5 games on PlayStation Network.

Master & Commander
Win a 5 player ranked Command Room game on PlayStation Network.

Silver

Capitalize This
Control 5 capitals.

Ironman
Defend a minimum of 5 attackers with 1 defender.

Demon Child
Achieve 10 super overkills.

Star Command
Earn 10 stars and trade them for 30 troops.

Glory Hunter
Win 10 ranked games on PlayStation Network.

Gold

5 Star General
Win a game on each map.

http://www.ps3trophies.org/game/risk-factions/trophies/