At the beginning of the game, you will have the choice between Human, Hero, and Super Hero difficulty. Select Super Hero and complete the game in order to unlock this trophy. Super Hero is The Amazing Spider-Man's Hard difficulty.
This trophy stacks with Peter Parker and Vigilante; completing the game on Hero will also give you the trophy for completing it on Human.
If you choose to play through on Super Hero first as I advised, you will probably find yourself playing much more stealthily than if you went for an easier difficulty. Try to take down as many villains as possible in the shadows, using to perform stealth takedowns, and don't be afraid to web retreat with if you find that your health is dropping too low. Most sections can be gotten through if you go slowly and carefully - rushing in is more hazardous, and of course, practice makes perfect.
If you play through on an easier difficulty first, Super Hero won't be half as difficult, since you'll know exactly what to expect in each level.
Potentially Glitched – It has been reported that people have completed the game on Super Hero and only received the trophy for completing it on Human. In order to avoid this, before starting a new game, go to the Options menu on the Main Menu and change the difficulty to Super Hero. Then select Super Hero when you are given the choice after starting a New Game. This should ensure you get all the trophies you deserve when you finish the game.