You start out under attack. Select everyone, and retreat to your base. Let the photon cannons take out any followers. Now you need to concentrate on building up your economy and upgrading your units. Set one of your probes to gathering minerals, while the other builds structures for you. Build two more forges, two robotics facilities, two more gateways and another stargate. While this is going on, be pumping out more probes. You need thirteen of them on minerals and three on gas. Build some extra photon cannons around the borders of your base. Namely, three on the eastern side, spread out near the upgrade buildings, two near the stargates and one extra at the entrance to your base. Research upgrades. You're going to want all 3 levels of all upgrades at the forge, all upgrades at the cybernetics facility, all reaver upgrades and the increased speed for your zealots. Do other things while these are in progress.
You are going to have to defend, a lot, before you can start on the offensive. Keep a balanced force of dragoons and zealots ready at all times to reinforce your position at the entrance. When you're done with economy-building you need to plan to expand. Again, make a good force of zealots and dragoons for this. You probably have spare gas and could build a templar archives to make archons. Three archons, six zealots and four dragoons should move on to the expansion area, which is east of your base. Build three photon cannons with a pylon on either side, as well as a nexus and assimilator. Six probes on minerals and three on gas here should suffice.
Now your economy should be very strong. Start building reavers, get a total of eight of them. While these are building you need to keep your forces stocked at either base to defend it. Scouts make good defenders here because they can't be damaged by reavers. As the upgrades continue to finish during this time, keep grabbing the next level. Keep your defense up and while that's going on start building up an army to assault the enemy. This is what I took, and I had some left over. 24 zealots, 12 dragoons, eight reavers and eight scouts. An observer should be present to detect enemies cloaked by the arbiters.
The reason not to really be building archons is because they are juicy targets for enemy reavers. Only build those if you have an excess of gas. If you have the ability, add some of those to your army. The dark templar (minus zeratul) can go wit the archons. When you have your army (it will take a while, considering the defensive obligations) head to the red base to the south of yours, on the western side of the map.
This will be a very tough battle because the red and possibly orange will both send reinforcements. Let the zealots run amok while you focus fire with your reavers and scouts. The scouts' primary objective will be to attack incoming reavers. They're quite tough, so en masse they should do a fine job at taking them out even under fire from other enemies. You'll want to destroy this entire base! When it's over you need to replenish your forces. I know, it sucks to lose momentum like this but there is another base to take down.
You see, destroying the heart of the conclave means taking down two of their bases, be it orange or red. I felt the red was easiest. Once the first red base is down, you'll want to go after the second, which is located past the bridges (see screen) that were east of the first base. Head north of there (use some hallucinated scouts via Tassadar to locate it). The base should be easier this time because they sent so many reinforcements. Still, you should try to duplicate the first force to be safe. When the second enemy nexus (I think that's what triggers it! Comments?) is destroyed the mission is complete.