Pretty simple trophy. Famine Mode is a cheat code collected as a secret in the campaign. Enabling this cheat will prevent health, ammo and armour drops from Demons.
I personally waited until after I had completed the campaign to do this. You don't have to but it makes life a little easier, even though this isn't particularly challenging. Simply select the first mission and hit square to enable the cheat (only available if you've collected it in game) . If you're using a save slot on a higher difficulty you can lower it in the pause menu once the mission starts to make things even easier. Doing this with everything you've obtained throughout your save slot makes the achievement pretty trivial. Demons will drop nothing upon death, with the exception of Chainsaw use, and regular ground pick ups are still available. Using grenades, flame belch and blood punch can save on ammo so be sure to use those when they're available. There is also a very useful Praetor Suit upgrade which grants a bit of ammo when using exploding barrels, so if you find yourself getting a little short be sure to destroy every barrel you see. Just progress through the level and the achievement will be yours.
Easy Peasy :)