Here is the full list of all 25 Resistance: Burning Skies trophies - there are 10 bronze, 7 silver, 7 gold and 1 platinum.
Successfully complete level 1
Successfully complete level 2
Upgrade a weapon in the Single Player Campaign
Upgrade both slots of a weapon in the Single Player Campaign
Kill 250 Chimera in the Single Player Campaign
Kill 50 Chimera with Riley's axe in the Single Player Campaign
Kill an Executioner
Kill the Abomination
Earn all trophies
Successfully complete level 6
Kill Gorrell
Kill the Leviathan
Complete one round of multiplayer
Kill any combination of 18 Impalers or Executioners
Kill 1000 Chimera in the Single Player Campaign
Successfully complete level 5
Kill 50 Chimera by detonating their heatstacks in the Single Player Campaign
Successfully complete level 4
Use the secondary fire of each weapon in the Single Player Campaign
Kill an enemy with every weapon in the Single Player Campaign
Upgrade all weapons in the Single Player Campaign
Kill 100 Chimera with headshots in the Single Player Campaign
Kill 500 Chimera in the Single Player Campaign
Successfully complete level 3
Kill an Impaler