This guide is for those who are currenty tearing their hair out trying to complete this achievement. After countless retries I finally found a build that works after reading through various forums etc. Although getting the achievement seems very difficult, it is deceptively very easy to get. This guide assumes you are just going for the achievement and nothing else. This is my first guide and I'll explain what I did as best as I can.
Start by building up your economy. Build as many scvs as you can, but only queue 2 scvs each time (reason being spamming units on the build queue just wastes resources and delays your other units). Ideally you want to have around 12-16 scvs on your mineral line and 2 scvs on your refinery.
Scrap the 2 bunkers on the lower ground and move your forces to higher gound. Fill up the 2 bunkers on the high ground. The single entrance allows to focus your defense in one direction and thus spend less resources on defense. You should have 8 marines standing around after you do this.
In the first 1-2 mins, while producing scvs, build a tech lab and a reactor on the respective barracks. You may produce a few marines/medics as long as you have the resources to spare. Soon you will be prompted to save the first group of marines on the right side of the map. Push out with the 8 initial marines and save them.
When you return to your base you can start getting an additional barracks + reactor. At this point you should have enough scvs to provide a steady economy. Start spamming marines and medics from the 3 barracks. However, as stated earlier, do not get overzealous and spam too many units on the build queue, as doing so may mean losing out the minerals and time to build supply depots and whatever else you need to get your production as steady as possible.
Focus on building up your army up till the 11-minute mark. You will want to build up your army AT LEAST until you reach 90-supply for your push to work. Once you have your army, try to push out ASAP around 11 mins. This is because, I find that once the timer reaches 10 mins and below the zerg will build up more units/defenses, making your job harder.
The zerg base being targeted is the one at the top left corner of the map containing 2 hatcheries. As you push out, use attack-move (press A and click) to clear out enemy forces you meet along the way. Now this is VERY important as this allows your marines to pick their own targets and kill the zerg groups you meet quickly. If you panic and spam-click randomly you will suffer more losses than necessary.
If you time your push right there will only be one spine crawler blocking your path. You should ONLY use focus-fire on the spine crawlers you meet. Take it out and move to the top left corner of the map outside the zerg base.
Once you reach the zerg base, try to lure some of the defending forces outside the base to make them easier to kill. At this point the defenses should include a few mutalisks/hydras/roaches + a reasonable amount of zerglings. Once you kill this initial group you may start to push in slowly and kill off the 2 spine crawlers. Once you focus down the 2 spine crawlers revert back to using attack-move while you hit the 2 hatcheries since the zerg will continue to throw units at you. However they should come in groups of 2-3 at a time and your marines have no difficulty killing them while hitting the hatcheries if you use attack-move. Now, while doing all this, DO NOT KILL THE DRONES!
Once the 2 hatcheries are down and you still got a good portion of your army left (I had at least half my army remaining after taking down the 2 hatcheries), well, congratulations, the hard part is over! Now, what do we do about the 2 remaining hatcheries? You DID leave some drones alive, didn't you? Well all you have to do is to wait for maybe 10 seconds outside the zerg base, and the drones should start respawning the 2 hatcheries. You CAN kill the hatcheries while they are still building so kill them while they still have low health! Volia! You have your achievement!
Once that is done what you do from that point onwards is your own decision. You get "The Best Defense" achievement instantly and do not have to complete the mission to get it. Anyways, this sums up my guide and I hope this helps! :)