Prototype

Save before every mission, and if you should die, quit to the main menu and reload your save so you can try the mission again. If you are patient, you should be able to get this the first time you play the game on medium. However, if you are having trouble, you can continue to play without getting this trophy right away, keep playing past the story missions until you have unlocked all of Alex's abilities, and then start a New Game + on Easy. This will carry over all of your abilities, putting you at a huge advantage and making this trophy feel like it should only be a bronze instead of a gold.
This trophy just doesn't count deaths in missions: if you die at any time before you finish the story, you lose your chance at this trophy unless you reload a save with 0 deaths. Your death count needs to be at 0 when you complete the final mission. However, only actual deaths, where the game tells you "ALEX IS DEAD", count against you. If you simply fail a mission, you can restart from the last checkpoint without penalty. This trophy will unlock after the credits and you have started playing again in Free Roam mode.
sulhaq has come up with an alternate way to keep your death count at 0 without quitting the game and having to start the entire mission over: "Get the Adrenaline Surge ability ASAP. Now, whenever you are about to die, instead of dying you will go into Critical Mass to give you one last shot at a Devastator. Instead of using a Devastator, just pause the game, reload the last checkpoint, and try again." To reload the last checkpoint from the pause menu, go to Quit -> Restore Checkpoint.

You can track your progress under the Statistics -> General page on the pause menu.