This achievement can only be earned in the Strategy Game, not the Campaign.
The concept behind this series of achievements is to not ever load a saved game while capturing five islands held by the enemy AI. Note that islands that are green on the strategic map are not occupied and do not count for this achievement. This also means that if you blow up the Command Center on one of the enemy islands, that island will not count for the achievement either since that turns the island to neutral.
Start a strategy game with the following settings: Captured Islands Count (between Half and Most), Enemy Strength (Weak) Startup Resources (High), Islands Output Rate (High), and Victory Conditions - Destroy Enemy Carrier (Required). You will start with most of the islands, and you need to make sure the enemy has five under his control. If he doesn't restart the scenario with your captured islands count slightly less. Provision up as desired and start off. The best approach is to sink his carrier first so that he doesn't go around capturing any of your islands. To do this build at least two torpedoes (in case you miss) and then hunt him down. You will get alerts telling you he is attacking an island, so just head there and intercept him. The most reliable way of killing him is to load up a Manta with a torpedo and drop it in straight from behind, then bring your carrier up behind him and lay into him with your plasma gun. If you want to launch missiles or rockets from your Mantas while the AI fires your plasma it will go somewhat faster. Once you have him sunk, you will get a message asking if you want to continue, and you need to say yes.
Now you must capture five of his islands without EVER loading a save game, so this means you need to do this in one sitting. If you load a save for any reason - including the game freezing on you - the whole process must be started over.
Once you have unlocked "Hardcore MK.III," capture any remaining islands including the AI's stockpile if you haven't already done so (for "5 Finger Discount") and all but one neutral (green) island. Save here and use this for the "Millionaire" and "Hammer and Nails" achievements. When you are done with those, capture the last neutral island to unlock "Conqueror of the Dead Zone" and end the game.