Main Objective: Collect 100 Eggs [0/100], Zagara Can't Collect 100 Eggs [0/100], Destroy Zagara's Base
Bonus Objective: Destroy Infested Command Centers [0/2]
Domination: Complete the "Domination" mission in the Heart of the Swarm campaign.
Poached Eggs: Hatch all 6 Baneling Nests in the "Domination" mission before collecting 100 eggs.
No Egg for You: Don't let Zagara collect an egg in the "Domination" mission on Normal difficulty.
Zagara will attack periodically. She's fairly weak, so just be aware of the timer.
You will have enough time to clear out the entire eastern area by your natural expansion. There aren't many units defending those areas and you will find banelings along the way. Collect the eggs and resources you find and start your expansion once this area is cleared.
There is an area just north of your base which has a few eggs and some resources. The only real threat is the Infestor towards the back, which will Fungal Growth your zerglings, killing them if they're caught. Head in first with Kerrigan to take care of the Infestor before sending in your zerglings.
Make sure you have a few banelings at your base for defense. A few banelings can easily kill a large group of attacking zerglings.
The first Infested Command center is just north of your base. One control group's worth of zerglings will be enough to take care of the defending zerg units.
There are a handful of eggs and resource directly to the east with minimal defenses.
There are enough eggs in the northwestern most area to complete the mission. Don't collect them until after you've hatched all 6 baneling nests for the Poached Eggs achievement. The majority of the defenses here are spine crawlers.
Once you collect enough eggs, it will begin the final phase of the mission which is to destroy Zagara's base. You'll be pleased to find a large group of banelings awaiting your command after the short in-game cutscene.
With this many banelings, you can easily a-move to victory. More will spawn by your base shortly. Keep in mind, the mission will not end until you've destroyed the majority of Zagara's buildings. Simply destroying her Hive and Hatcheries will not be enough.