This achievement can only be obtained on Claire's game, as Leon's final boss is Super Tyrant, not Stage 4 G.
At the last typewriter (after you go down the central elevator in the lab), open the item box. Keep your pistol for the enemies along the way to the boss, and fill the rest of your inventory with as many knives and healing items (preferrably G+R+B herb mixes) as you can hold, leaving two item spaces free for the fuse and minigun. When you enter the next area, the 10 minute timer starts. Go down the lift and quickly run through before the Ivys stand up. Continue down the ladder in the next room. Examine the door and Sherry will climb through the vent. Shoot at the bulbs on the Ivys to knock them down and/or keep them at bay. When Sherry says "I got it!" you can run through the door. You will come to a corridor with a zombie on fire blocking the door, shoot him once in the head to put him down. Keep running along the walkway and into the next area with the train. If you have any more healing items or knives in the item box, quickly swap your pistol for one of these. Get on the train and grab the fuse, then run into the control room and put it in, and grab the minigun after the cutscene and run back out to the train to start the fight.
When the boss fight starts, wait for him to come down to the ground, then get in close and start swiping with the knife, aim for the eyes. (I recommend setting your graphics to the Performance Priority preset, as your knife will do more damage the higher your framerate is.) He will knock you down several times, just keep healing when you hit danger and knife away. When he jumps up on the wall you can use the minigun to keep doing damage until he comes back down. The eyes will go from white, to yellow, to red to indicate his health. Once he starts oozing and crawling on the ground, you've won. He'll die after crawling around for 30 seconds and if you have more than 4 minutes left on the timer, you'll get the achievement.