Welcome to the Jungle

This is another one of those achievements that were pretty hard. Mostly it's because you are misdirected by the Goliaths you get for the mission and although they can be used to succeed in the normal mission they are too weak for their cost to be used effectively. There are a few ways to prevent all the capping and the most direct one I could think of was to learn the sequence and the timing for all attacks and capping, this however is very time consuming and you will have to play the map several times to get to know it exactly.

What I figured, and I also got this suggestion from several other sources, was to mass out infantry. For this you would want all the marine and medic upgrades; Combat Shields (marine hp increased by 10), StimPack (marine attack and movement speed increased by 50% for 12 seconds at the cost of 10hp), Advanced Medic Facilities (allows medics to be produced without the tech-lab addon) and Stabilizer Med-packs (medics heals 25% faster and for 33% less energy, basically making each medic close to twice as efficient). Start producing SCVs, add you a reactor to your barracks and start producing marines and medics. You don't actually have to push out and get any gas until you have secured the whole map but you can if you want to finish up faster.

You need to push out to block the first cap in the west side though, and eventually in the east. I basically left my western force in place and trained a complete new force to push out east after the first assault on the base. Build a second barracks and attach a reactor to that one too, you basically want each of your forces to have a 3:1 composition of marines to medics and a force of perhaps 10 marines and 4 medics can take down most early assaults. As you have a pretty high production of infantry you can easily reinforce both your troops as well as building up a third force in your base. This third force should push out and take down the Protoss base to the immediate north of your own base and set up an expansion there for extra income. Also make sure to get the upgrades for your infantry.

You don't actually have to go through all this as you can probably cap all seven of your gas canisters by now but I found it entertaining, as well as safer, to take control of the whole map before I even started capping anything. From this point you will have a lot more income also so you can build a third barracks, on which I put a tech-lab to make marauders to make an easier job with the ground units and towers as well as producing Goliaths. You can easily push out further and basically make static defenses outside the Protoss base entrances, there are two, one in the west and one in the east, but make sure you take out all Protoss buildings and units outside the base, especially the pylons and warp prisms so they can't warp in more units behind your lines.

When your forces are strong enough you can pretty much get out seven SCV's and simultaneously cap all seven 7. You can also muster up a big enough force of marines, medics, goliaths and marauders to destroy the protoss bases for the feat of strength achievement "You'ze So Crazy!".