This particular trophy doesn't require you to complete the game in any particular time. You simply need to start the game by going through the Speedrun option on the main menu.
Keep in mind that playing in Speedrunner limits you to only one save, so you can't copy or backup your save file. It will also skip any cutscenes and dialog in the game, including information screens when you pick up new weapons, tools, or upgrades.
All other trophies should be unlockable in this mode (I unlocked several 100% trophies in speedrun), but I also went for the Mostly Invincible trophy as well, since the time stamp gave me a good indication of whether progress was saved after I died.