Your first task on Char is to regain control of Zagara's brood. The brood mother has refused to accept Kerrigan as her superior, and is going to prevent any other work on Char until she is destroyed, controlled, or removed from the planet. This first mission offers fairly good rewards- five levels for Kerrigan on completion, a chance at two further levels for the psion, and access to the baneling mutation for zerglings.
As the mission begins, the main goal is revealed- Char holds numerous Zerg eggs that have been stored for later use. The rogue brood mother is here to collect as many of them as she can and hatch them in the birthing pools found in the acid marsh. Kerrigan intends to prove superiority in the most direct way available- collect a larger number of eggs and hatch a larger army with which to overtake Zagara's forces.
You are given the usual basic base, with the addition of a baneling nest- which allows you to transform your zerglings into their suicidal baneling form. Zagara attacks your base immediately- but is interrupted and driven off by a cluster of feral banelings that erupt from the ground on the border of your controlled area.
Once she is fended off, a number of eggs are revealed to you- mostly to the North, but there are a noteworthy number off to the East as well. Several clusters of feral Zerg units are also reavealed by this, clustered on creep around the egges to the Northwest, North, and Northeast.
Begin by building up your capacity to create forces- as useful as the eggs are and as much as they are key, you can't do anything about them if you haven't got the units or resources to sustain the necessary outgoing expedition to recover them.
Before long, Kerrigan and Eisha will note the presence of two infested command centers to the North- one near the center-East of the map, and one closer to the central North. These buildings are apparently of great import to the local infested Terrans and other Zerg, and each one that you destroy will grant Kerrigan an additional level. You will also start finding baneling egg clusters as you wander around, each hatching into 8-10 banelings when Kerrigan is close enough to it- use these to bolster your forces for a faster sweep around the map. It shouldn't take long before you've built up a decent assault force around Sarah, and can simply turn your entire attention to gathering eggs.