The Amazing Charge attack is unlocked in the 5th section of upgrades going down (seperated by red lines). It's the first attack on that list after completing enough challenges to unlock that section. It costs 7,000 essence. This move can only be used while playing as Amazing Spider-Man, which is the basic red and blue suited Spider-Man. On any level you use that character, press and hold then release to launch the attack. You must hold it down long enough to see the web circling around Spider-Man's fist. After playing more of the game, you'll unlock stronger versions of this attack (level 2 and 3) which will help you get kills faster then grinding it out early on without them. Get 50 kills with this and the achievement will unlock. You can grind these on the first level on easy if you want. Restarting your checkpoint is okay, as it will still save the stats you earned.
You must purchase this move from the upgrade menu and can only use it as The Amazing Spider-Man. Hold X down and when the web is covering your hand, let go at an enemy. It's best to lock on to them. If you restart a checkpoint all of the ones you killed will still count.