NOTE: You'll get this mission if you choose to help Ariel Hanson and protect the Haven settlers from the Protoss
This mission can be tricky if you're not prepared. The main objective isn't too hard, but completing the secondary one is difficult unless you start producing Vikings ASAP.
1. Build Vikings, assemble a ground force
As soon as you've got your SCVs going build a reactor on your Starport and your Barracks. Start building 4-5 Vikings as soon as you can afford to while pumping out marines to assault the Nexuses. Build an Armory and an Engineering Bay ASAP also, upgrading ship and infantry weapons first. You likely won't need to build a Tech Lab as long as you keep your 2 medics alive, and you can use the mercenaries if you need Mauraders/Firebats.
2. Assault the first Nexus
Once you've got a squad of marines about 7-10 deep with the 2 medics you were given, assault the first nexus point. It's lightly defended and shouldn't be a problem.
3. (Achievement) Take down the first Terror Fleet
Around this time you'll be given the secondary objective of defending the refugees from the Terror Fleet, a marauding group of Carriers that attack the settlements. Fortunately the Vikings fair pretty well against these ships and you’ll only need 5-6 to shut down the first wave of the Terror Fleet.
4. Assault the second Nexus
Continue building marines and with a squad of around 15, assault the second nexus point. This one is slightly more defended as there is a huge Protoss Colossus here that can make short work of your marines. Use your stimpacks and focus fire it first before clearing out the Stalkers and Proton Cannons. After clearing out the base, leave this squad in position as you build more troops for the assault on the final Nexus.
5. (Achievement) Eleminate the second Terror Fleet
The second Terror Fleet will appear; hopefully you've built some more Vikings or repaired your existing ones. 6-7 Vikings will do just fine, especially if you've upgraded the ship plating and weapons at the armory.
Your base may be periodically attacked while you're assaulting the Nexuses and the terror fleet so it’s good to build a single bunker with 4 marines and an SCV just in case. A walker will assault your base near the end of the match, but he'll be no match for the bunker.
6. Assault the final Nexus
The final Nexus is much more heavily defended that the other two and you should bring your Vikings along to support the ground troops as there is a squad of ships that will need to be handled. It's good to supplement your marine/medic ground horde with some mercenaries before the final push, specifically the Mauraders, as they'll add a much needed punch. Assault the final Nexus, taking out the Carriers first, destroy it and the Purifier Mothership's shields will fall.
7. (Achievement) Eleminate the final Terror Fleet
The third Terror Fleet will appear at this point which can be problematic if your Vikings are still injured from the assault on the 3rd Nexus. Take some time to have your SCVs repair them or build additional Vikings; you can lag a little before saving them if necessary to make the repairs.
8. Destroy the Purifier
The Purifier Mothership is usually escorted by a few air units, but the Vikings can make short work of them. Make sure to supplement your attack on the Mothership with some marines, as the Vikings take heavy damage from the Mothership and will likely need assistance to bring it down. Use the Marine's stimpacks to boost their attack speed as they will usually be killed in one hit by the Purifier's laser anyway so the health loss is not an issue. Defeat the Purifier to end the mission.