This is one of the harder missions, but if you're a tad cheap, and patient, you can beat it. The method I used is as follows: Build up ten drones to gather minerals at the starting location. When they're done, use another to start an extractor. Get three drones working on gas. You will probably need an overlord somewhere during that. Now, build another hatchery as close to the extractor as possible to swiften both your gathering of that resource and unit production. Take your four air units directly south of your base to the area in the screenshot (note that screen is after I'd destroyed the protoss) and destroy the protoss there. Focus on the dragoon tanks first, as they'll be most bothersome to you. Clean that entire plateau up, then upgrade your hatchery to a lair and research ventral sacs.

Work on building up your base during the time that your hatchery is upgrading/getting the ventral sacs upgrade. Get two evolution chambers, two spires and a queen's nest. You'll now build a third hatchery near the extractor on that plateau. Place it as close as possible, and put three drones gathering gas there as well. Get a few drones working minerals there, too. You'll now have plenty of cash rolling in and can upgrade your units. Get the missile/carapace upgrades at the evolution chambers and both upgrades at the spires. When you're ready, build a Hive to allow you to upgrade to the greater spire, thus allowing you to build guardians. Another useful upgrade is for queens - ensnare. This will be great later in the mission. During all that, get three queens.

If your forces are dying from the protoss attacking, reinforce with hydralisks. These are going to be your primary units for this mission. Get every upgrade you can for them. In the meantime, get a total of ten guardians. You started with two, as well as two mutalisks. Here's how your forces should be: two groups of 12 hydralisks, one group of 10 guardians and three queens, each in their own groups. The queens will be used to utilize the ensnare ability on groups of dragoons at the protoss base. They'll double as scouts, too, so don't worry about wasting them once their energy is spent. Finally, hotkey all three of your hatcheries into its own group so you can perform the place rally point/mass reinforcements trick.

Now, don't let your base be undefended as you do this. Keep at least 6 hydralisks and Kerrigan home to do the defending. Take your 24 hydralisks, 10 guardians and 3 queens south of your base. You'll see the location in the screenshot to the right - that is your ultimate destination. Notice how I used the ensnare ability on the dragoons. That's very useful, and as soon as the queen was clear my forces managed to romp that group. You're going to have to be more tactical than that in your approach to this mission, but it's a good start.

Use your guardians to slowly advance (queens can scout) and attack anything anti-air. Only bring the hydralisks in if there are a number of things firing at your guardians. Use them to help destroy the anti-air capable enemies. Just don't let a single zealot sit there and slaughter your hydralisks because you're not paying attention! If you see a reaver as you advance back off your hydralisks. Focus the guardians on it so they can return to battle. The scarabs those reavers fire will slaughter you. Trust me, you'll see what I mean. As you progress up the hill to the east and enter the main protoss base, set your rally point of all three hatcheries to the area you're fighting in. Do it like this: 8, right click, 9, right click etc. Then hit 8 s m, 9 s m, 0 s m and mutalisks will be on the way to help you after they're hatched. Keep your eye out for reavers, focus on dragoons with your guardians and fire at photon cannons with guardians when you're out of their range and you'll soon be victorious. If you somehow fail, but deal a lot of damage get back into the action, fast! Don't let the protoss recover.