Although a lot more difficult than the previous mission, this one is still pretty easy. You start out in a well-established base with Raynor, 4 Marines, and 2 SCVs. Send the SCVs to gather minerals immediately while building more SCVs and Marines. When you have 5 or more SCVs send one to build a Refinery on the Vespene Geyser north of your Command Center and send another to help it gather Vespene Gas.
When you have 11 Marines, group them with Raynor and move northeast outside your base. Use Raynor to slaughter the Zerglings in the dead end. Then, repair Raynor with an SCV and send your Marines and Raynor up the ramp north of your base. You will encounter 5 Zerglings and a Creep Colony. After you vaporize them, move up into the purple Terran base. You will get 2 SCVs, a handful of Firebats, and some buildings, including a unfamiliar one: the Academy. Send the SCVs to gather minerals, and research the U-238 Shells and Stim Pack. You can also upgrade Infantry Weaponry and Armor in the Engineering Bay at your original base.
Rest at your new base while building more Firebats and Marines. When you have 12 Firebats and 12 Marines, move the Firebats southeast out of your new base and destroy the Zerg encampment there. Then, group your units tightly and move south into the canyon. When you clear the area of Zerg, proceed northeast onto a cliff. There, on the left and right, are two Zerg encampments. Destroy them one at a time. Then, go all the way northeast and destroy the Infested Command Center to beat the mission.