To unlock the Deep and Hard trophy in Call of Duty: Modern Warfare Remastered, you must complete the single-player campaign on either Hardened or Veteran difficulty.
Playing on Veteran difficulty isn't too hard, but there is a glitch that you can use to complete each mission on Veteran difficulty whilst playing the vast majority of the time on recruit.
The way the glitch works, is you play the mission on recruit difficulty until you get to the final checkpoint of the level. Return to the main start screen, go to Mission Select, choose the mission you have just been playing and select Veteran difficulty. As soon as the mission begins to load, quit to your console dashboard and close the game completely. Restart the game, and go choose Resume Game, and your progress will be saved at the last checkpoint, but you will now be on Veteran difficulty.
You can see a video of the glitch working below (credit to It's Ya Boy on YouTube).
You can find a list of the last checkpoint for each level below:
*** Spoiler - click to reveal ***Crew Expendable: After the explosion and your teammate helps you up and you start sprinting to exit the ship.
Blackout: After the power is cut and you enter the house to retrieve the objective.
Charlie Don't Surf: Just after the big room gun fight with all the TVs and you head up to the roof to breach the last door for the last 5 TVs.
The Bog: After blowing up the AA gunner.
The Hunted: Just after shooting down the chopper with the javelin.
Deaths From Above: When you hear the commander suggest that you use the 25mm, just before noting friendly choppers are in the area.
War Pig: Just after a gunfight and a briefing you drop down to the tank and head towards the LZ.
Shock And Awe: After rescuing the downed chopper pilot and placing them on the evac chopper.
Safe House: After Price throws the flashbang in the door and closes it.
All Ghillied Up: When you encounter the dog on the leash.
One Shot One Kill: After placing Macmillan in his final sniping spot by the Ferris wheel.
Heat: After dropping down to the gas station and boarding the LZ chopper.
Sins Of The Father: Before the last set of stairs going to the roof. If you reach the rooftop stop and perform the glitch here.
Ultimatum: After you fend off the two trucks and go and pass through the gates and the sirens start sounding.
All In: After blowing up the final tank before your team starts cutting open the vents.
No Fighting In The War Room: After you get off the elevator.
Game Over: When you're lying on the ground there will be two before Price slides you a pistol. When the chopper blows up and when he slides you the pistol. These are both usable.