To make this Starcraft 2 guide effective against Zerg enemies, your highest priority should be taking down their Hatchery as they will try to bring out an Overseer when you attack them with Dark Templars. If you are able to sneak into their base without being noticed, 4 Dark Templars are usually enough to take down a Lair before an Overseer is produced.

Do note that Dark Templars alone cannot hope to defeat a Zerg player as it is extremely easy for them to obtain an Overseer (detector). However, f you take them by surprise you can kill many Drones and even a Hatchery.

Dark Templars can be used to hurt the Zerg economy once, but to finish them off, you need your regular units. Generally, it is not recommended to do a Dark Templar rush against Zergs. Just use your Dark Templars to hurt their economy and supply count by taking out their Overlords with the Phoenix. Focus on Overseers first, then on Overlords.

While the enemy pays attention to the Phoenix, sneak in your Dark Templars and destroy any anti-air structure or units you find like Hydralisks or Queens. Then use the Phoenix to take out Overseers that may have begun forming.