Main Objective: Destroy the Eradicator
Bonus Objective: Release Zerglings from Pens [0/3]
Lab Rat: Complete the "Lab Rat" mission in the Heart of the Swarm campaign.
Mighty Mouse: Destroy 35 Sentry Bots in the "Lab Rat" mission.
Rat Race: Complete the "Lab Rat" mission in less than 10 minutes on Normal difficulty.
Even though this is brutal difficulty, the prologue is essentially a tutorial level. There's nothing special that you need to do here aside from a-moving to victory. Keep in mind, there are no hidden items in this level, so you can go through it quickly to get the Rat Race achievement.
There are 3 pens which are highlighted on the mini-map. Destroying these will give you even more zerglings. Make sure to have an adequate number of drones mining minerals so you can keep up with zergling production.
Another objective will appear to destroy the 3 gas turrets in this room. If you lose all of your zerglings, it's not a problem. All it will do is make the Prologue even longer. Rally your hatchery right outside of this room to help speed things up.
You can choose to keep sending in zerglings to destroy the factories if you like because they do show up on the score screen. They don't appear to be tied to any achievement, so if you go straight for the Eradicator, you should have nothing to worry about.