Teplar's Return. Requires you to complete the mission on any difficulty level.
Outrunner. Don't let the psionic energy chasing you kill any of your units during the third part of the mission. A very simple objective, you must only keep going forward and not stopping. Or you can stop but only for short breaks when the energy is far away behind.
Master Control. Take over at least 15 enemy units using Karax's ability while playing on normal difficulty level. Move using the whole army in the second part of the mission. Whenever you notice the enemy, give zealots and archons order to attack it. At the same time let Karax use his abilities. If the enemy has two units that can be taken over, retreat after taking the first one and wait for Karax's ability to regenerate.
Silent but Deadly. Don't let Vorazun receive any damage when playing on high difficulty level. Walk into the detector's range as rarely as possible and let Alarak carry the burden of most battles. Optionally you can lure the enemies outside the detector's range where Vorazun will be able to fight without worrying about being hurt.