1. Quickly bolster your defenses
You need to act quickly here as the Zerg will be relentlessly attacking you during this mission. Fortunately you start off with a full base and a lot of units already built. Depending on whether you destroyed the Nydus Canals or the Air Unit platform, you'll want to build extra Siege Tanks/anti-infantryof Missile Turrets/anti-air. Make sure you've got 3 bunkers on each side of your base and use any researched anti-Zerg tech, like the psi-disruptor or Perdition turret, if you have it.
2. Begin building Battlecruisers and Vikings/Wraiths
Battlecruisers are incredibly helpful in this mission as their abilities are already researched, and you'll want to do as much damage to Kerrigan as quickly as possible. Left unchecked, she can run through your defenses in an instant, so try and cut her off with your Battlecruisers. TheirYamato gun shots will quickly force her to retreat but make sure to bring some other aerialunits to soak up her 1 hit kill move.
3. Beat back Kerrigan
Not long after you've defeated the first few waves of enemies you'll be given warning that Kerrigan herself is approaching. She only approaches from your base's two forward points, and she appears on the map, so you'll have a moment to gather your forces to take her out. As previously mentioned, try and just spamBattlecruisers at her, as it's easier justto queue up a bunch of ships than to have to rebuild bunker and turret positions.
4.Rebuild your defenses
You'll likely lose a bunker or some units while stopping Kerrigan or managing the myriad tasks you have to complete during this mission, so make sure to have SCVs ready to go at all your major choke points to repair and rebuild. A small SCV brigade to repair Battlecruisers is also helpful.
5. Repel the Leviathan (Optional)
The Leviathan only appears if you destroyed the Nydus Canal, but can quickly crush your base if left unchecked. You'll get a warning when he makes his approach, so make sure to gather your Vikings and Battlecruisers to take him out. Yamato Gun/Vikingfocus fire will usually do the trick.
6. Don't be afraid to use the Artifact
The Protoss Artifact recharges every couple of minutes, and unless you're going for the achievement, don't worry about using it when things get hairy. You'll usually be given a warning when large waves are incoming, so feel free to pop it if your defenses are wavering.
7. Be prepared for drop pods/Nydus Canals
As the mission marches on there's a good chance the Zerg forces will start deploying behind your defensive line. It's easy to not notice this since you'll constantly be getting "Under attack" messages. Pay particular attention to the area around your Mercenary Facility as the Zerg seem very fond of deploying here. Also pay close attention to Brood Lords, who can attack your positions while remaining just out of range of bunkers and turrets, you'll need Vikings or Wraiths to clear them out quickly.