You need to complete the campaign to unlock Ultra-Nightmare difficulty.
You'll only need to complete the first mission on Ultra-Nightmare difficulty to obtain this trophy. Here's a few quick details on what Ultra-Nightmare difficulty is:

• There are no checkpoints, if you die you begin from the beginning of the game.
• Enemies deal a lot of damage.
• Glory Kills give very little health back.

Tips:

• Fight one enemy at the time, lure them to you.
• Use the charged beam on the pistol, it helps a lot.
• As soon as you find the Field Drone acquire the Explosive Rounds.

For a video guide on getting through this mission on Ultra-Nightmare, the video below shows a glitch that starts around the 2:30 mark which allows you to bypass much of the level, making this trophy a breeze.

A Toe Into Madness Video Guide (The Easy Way)

Credit to IceDoesntHelp for the video.