There are certain rules you need to observe and follow on using Dark Templars rush to win against Zergs.
Destroy the Hatchery so as not to let them create an Overseer
Use at least 3 or more Dark Templars to overcome Lair before an Overseer exists.
Overseer detects Dark Templars, so using them alone may not overthrow Zergs. Conduct a surprise attack and kill as many Drones as you can and destroy the Hatchery.
Dark Templars need assistance from your regular units, they will certainly adds up to bring down Zergs economy.
The main idea is to use Dark Templars to bring down Zergs supplies by destroying their Overlords with the help of Phoenix.
Focus on Overseers first, then on Overlords.
While Phoenix is doing it's attack use the Dark Templars to kill any anti-air structure like Hydralisks or Queens.
Finally use Phoenix to bring down Overseers that may have begun forming.