To get this you must beat the game in under three and a half hours. What I did to help me out was have a speed run video up on my laptop and letting it run as I'm playing so i know when to drop what items where and when to swap characters. Once you beat the game in under three and a half hours this will unlock the submachine gun(with limited ammo) and unlimited rocket launcher thus making the other smaller achievements easier to get. While the video itself is completed in 1hr 50min my run was done in 2hrs and 37min even with a few deaths and reloading saves and also rewinding the video to go back to a few things. This give you plenty of buffer time to make a few mistakes and what not. I did my run through on normal because like RE1 if you beat the game on normal or higher in under 3hrs you unlocked the rocket launcher.
Also once the Unlimited Launcher has been unlocked you can then use it in Wesker Mode
NOTE: If you pause the game the timer will still keep going.
EDIT: As Mich Hemsworth noted you can press the home/guide button
to dashboard on the controller and it will pause your game and also freezing the game/game timer allowing you to take a break from it.
Here's the Video I used: