This achievement is not as difficult as it may appear to be by its ratio.
First off, I recommend doing it on Easy difficulty because it would obviously make things, ahem, easier.
As for the first group of enemies and most of the first part of the level, just run, run, run. On Easy, the enemies don't seem to be too hell bent on killing you as they would be on a harder difficulty, so use that to your advantage.
Eventually you'll come up to a door which will remain locked until you clear out all the enemies. I recommend picking up a Scattershot (Promethean Shotgun) and Lightrifle (Promethean Precision weapon) just because it'll make things so much easier.
Use the Scattershot on Knights and Elites, and use the Lightrifle on the smaller enemies like Grunts and Crawlers. Obviously, only do so if you can't just sprint past them, which is the recommended approach.
When you reach a section which will require you to destroy three cores, you should try skipping any pointless fights and target the cores exclusively. After destroying all three, simply run up the main stracture onto the bridge and continue on running.
Another thing worth noting, at some point you'll see a scripted event in which a Knight kills two Elites driving Ghosts. You should quickly kill the Knight (or avoid it) and get into a Ghost. From there on, just use the Ghost to skip any enemies in your way. You'll soon come up to another scenario where you'll once again be tasked with destroying three cores. Needless to say, use the Ghost to your advantage and focus once again on the cores, ignoring the enemies. Once you're done with the cores, get to the main stracture, exit the Ghost and run up the bridge once again.
Eventually, you'll get to the point where you have two big doors with switches and a Covenant Phantom dropping reinforcements (Cortana will say something like "why are the Covenant interested in a relay station"). You should quickly run towards the switch, then enter. You'll see large groups of enemies, once again run past them. Keep running until you reach a device you are supposed to interact with and then you'll have a cutscene. Woah, we're not done yet! This level does drag forever, doesn't it?
Anyway, you'll get up, and again, run up to the point you'll go through a portal and there will be plenty of Ghosts for you to get in. Choose whichever you fancy and get going. From this point on, it's pretty linear, so just continue on until you reach the end of the level.
The par time is 20 minutes but I managed to do it in 15:05 on Easy, and I'm by no means a pro speed runner. Basically, you shouldn't struggle too much if you follow a few of my tips. By the end of the level Master Chief can qualify for a marathon with all the running he (and you) did, and you'll have yourself another achievement.
Sorry for the wall of text, I was trying to be as thorough as possible and at the same time avoid spoiling the story for any newcomers whom are unfamiliar with the game.
Good luck and thanks for reading!