The Xel´Naga created the physically perfect Protoss, but they felt it wasnt enough and moved to another planet. Where they found a simple life form that survived in a hostile ambient against all odds and decided to create a very adaptable race, with a perfection of spirit in mind, they created a large creature that could control that race, and called it the Overmind.
The Overmind learned at an amazing pace and quickly began to adapt the Zerg to its enviroment. Then he started to see the Xel´Naga as rivals and tryied to kill them, so they were forced to flee from that planet, with some casualties. Then the Overmind decided to look after that perfect body the Xel´Naga said they created, in order to assimilate its genetics.
It did assimilate a unnintelligent life form that could survive in space and started forming the Swarm. Conquering, killing and assimilating those worthy. When the Overmind created the Cerebrates (a very intelligent creature, with different personalities, but still under his control) the Zerg became an incredible killing force.
And one day the Overmind lost control over a space drone, searching for the Protoss and decided to investigate that energy. That space drone entered the Terran Confederacy, a small cluster of humans planets and colonies. The Protoss was also nearby and when they noticed Creep Colonies forming around those humans they decided to act and kill that race, even if that meant to kill those humans.
While on Char, Tassadar (a protoss commander sent to exterminate the Zerg) found the dark templars ruled by Zeratul, and with their dark power they did kill a Cerebrate. The others Cerebrates were terrified, since they thought they could not be killed, but the Overmind was happy, for moments before they killed that Cerebrate, the Zeratul was forced to make mental contact with it, revealing the location of Aiur.
Then the Overmind decided to take almost the entire swarm to Aiur, in order to conquer that old dream. All its most important cerebrates were taken to Aiur, but Kerrigan (a Cerebrate created with human DNA) was ordered to stay on Char and command their support Zergs. The Overmind was conquering and defeating army after army, since before the invasion the Overmind used almost all his power to teleport his enemies fleet and himself to Aiur.
Tassadar then sacrificed himself, using dark magic, psychic energies and the core of his flagship, creating a big explosion that killed the Overmind, but instead of winning the war, it set the Zerg loose, and the swarm enraged forcing all protoss to leave their home planet. All Cerebrates became independant and began to fight against each other to control the Zerg. A few of them decided to try to recriate the Overmind and they would be succesfull, if Kerrigan, now calling herself Queen of Blades, wasnt manipulating Terrans and Protoss to kill each one of them, and later, assume control of the Swarm, what she did at the end. At least of the Zerg she knew, since the Overmind was the only one to really know where and how many creatures he had under control.
A few early tips:
Zerg is very macro dependant race. You need quantities, to learn when and how to harass and is one race that can expand faster.