I have been refining the specifics missions and for now I'll give the run down on my planned Zerg Campaign, again spoilers ahead.
Mission 1 - Lab Rat: Alpha Squadron and Moebius Corp. attempt to assassinate Kerrigan on Char and fail. Introduction of Warfield and Tosh.
Mission 2 - Supreme: Kerrigan awakens the Cerebrate from StarCraft 1, named Zurvan, to lead the Swarm against Mengsk and who Abathur has replaced all traces of the Overminds dna and replaced it with Protoss genetic material, turning him into a Pseudo-Hybrid, Broodmother Zagara objects and Kerrigan pits them against each other.
Mission 3 - Planetfall: The Swarm attack Mar Sara, which is under the control of the Kel-Morian Combine, introduces Mira Han as a leader in that faction.
Mission 4 - Harvest of Screams: Same concept as in HotS, Broodmother was sent to Kaldir and killed, Broodmother Zagara goes to investigate and destroys a Protoss colony.
Mission 5 - With Friends Like These: Kerrigan attack a Terran settlement built inside an asteroid field, which unknown to her is being managed by the Magistrate, hilarity ensues.
Mission 6 - Death From Above: The fleet that was assigned to the Kaldir colony finds the Zerg still in the aforementioned asteroid field, and attempt to aid the Terrans fight the Zerg.
Mission 7 - Old Soldiers: The Swarm enters Dominion space and attacks a highly defended security outpost, Skygierr Station, which houses the bulk of Alpha Squadron and General Warfield's main base, Warfield escapes the stations destruction on his personal Battlecruiser, but not without a few unwanted guests.
Mission 8 - Infestation: Zerg infest Warfield ship, turning the majority of the crew into infested Terrans.
Mission 9 - Enemy Within: A renegade Zerg Brood on Tarsonis attacks Kerrigan brood, which leads her to believe that Mengsk has developed a means to control the Zerg, similar to the U.E.D.
Mission 10 - Fire In the Sky: The Swarm attacks Korhals orbital defense platform and the Korhal Defense Fleet .
Mission 11 - Domination: Basically a combination of the final two missions in HotS, Kerrigan confronts Mengsk, introduction of Valerian.
Mission 12 - Hand of Darkness: Kerrigan senses a psionic emination on a barren planet, believing it's Valerian with a new Psi Disruptor she attacks, finding Tosh and the Moebius Corp. Kerrigan is knocked out by a Xel'Naga device, her fate is unknown.
And that is my version of Heart of the Swarm.