I played through this mission many times seeking a better way to complete it, but this was the final method I settled on. It's unsatisfactory in that I hate rushes, or any instance of using a single unit type to complete a mission. Still, you are at a disadvantage of sorts in this mission due to the fact that the Terrans have many siege tanks set up, which would hamper your ground forces. You can avoid that nuisance with the following method.

Hotkey all your starting military and keep them near your drones for the start, so they can protect them. Build up eight drones harvesting minerals, then build a drone and have it on standby. Use your mutalisks and hydralisks to go up the ramp to the north of your base. Keep them together to avoid casualties, as you'll discover some terrans guarding a mineral deposit and vespene geyser. Destroy the terrans, then use your mutalisks to scout out the area south of there. Make sure there are no terran leftovers that will attack the Chrysalis or your workers, then set your group to guard the area in the screenshot.

Use your drone to build a hatchery near the minerals you discovered. You're going to want to pump out some drones (stop for overlords if you need them) to build an extractor at your first base, and put four to gathering vespene. Do this while your new base is constructed and your forces protect it. When the new hatchery is done, start an extractor and put four more drones on vespene at the new geyser. While it's being built, morph a drone into a sunken colony at the location in the above screenshot. You're going to end up with eight drones on each of the hatcheries collecting minerals and four on each extractor gathering gas.

Now how do you spend all those resources? You have a lot coming in. Morph one of your hatcheries into a lair and, when it's done, build two spires. Each spire will research a different upgrade while you build mutalisks. You're going to build a lot of mutalisks. While the spires are being built, make overlords at each of your hatcheries to make your support over 70. It should be 74. This is a good number, but you can go over a bit if you like. It will only make it easier on you. Build a total of eighteen mutalisks. Yes, eighteen.

This is the tricky part. Take your mutalisks directly north of your base. You should have one group of 12 and one group of six. Hotkey your lairs into 9 and 0 so that you can quickly switch rally points (right click on the ground) and produce more mutalisks. When you get toward the top of the map (see screen) you'll encounter the first of the terrans. Destroy the bunker and siege tank here. Set your two rally points of the hatcheries to this location.

If you can't build more mutalisks (this is cool: 9, s, m, 0, s, m will quickly make more mutalisks which will fly to your rally point) don't worry. If your forces start dying as you attack to the west, start building more mutalisks. It's important that you're quick with your attacks here. Focus the mutalisks on anything that is anti-air as you proceed west. This includes marines, wraiths, missile turrets and bunkers. Don't worry about siege tanks yet. If your forces start dying before reinforcements arrive, move them to the area I told you to set your rally point to. Hotkey the two groups again and continue smashing into the terran base. It extends to the west, then goes to the south some. When you've destroyed almost all the terran buildings, general Duke will pull his forces out.

Like I said, I'm not usually into winning battles like this but I was frustrated with the siege tanks and felt this was the best option. This method is pretty foolproof. If you fail, you're probably going a bit too slow and the terrans have built up too many forces. You'll get better with focus-firing frantically in battle as you play this mission, and it will be a good learning experience.