If you have been faithfully following the Main Quest walkthru that I've laid out, the Archcanon is the next stop along the path, albeit a very short one.
After completing the Cavern of the Incarnate quest, any of a number of NPCs will start telling you that you need to see the Archcanon (see the Hiccup page for why you should not have done this):
Now is the time to follow up on that.
Assuming that you have not picked the lock on the door to his office or snuck in through a back door, you will need a key and you get it from Danso Indules in the big tunnel on top of the Temple Canton. Indules will not give you the key to the Archcanon's office until you have completed the Fourth and Fifth trials (been named Nerevarine by all four tribes and Hortator by all three Great Houses).
Open the door to the Archcanon's office (it's through the left door of the High Fane) and talk to the Archcanon about an audience.
The conversation can be short and to the point: Vivec wants to talk to you. Here's the key. Go.
In theory (meaning this is the official word from Bethesda), the Archcanon will hand over the key to the Palace at any time after you have completed the Third Trial (retrieved Moon-and-Star), have an overall rep of at least 50 and are at least 20th level. I don't see anything in the Archcanon's dialogue options or scripts to substantiate the last two. I've tried at 24th level with an overall rep of 50, but no Moon-and-Star and got nowhere. But as soon as I fetched the ring, I was in.
If you have questions about the Tribunal Temple or any of the Saints, Archcanon Saryoni (noted author of "Saryoni's Sermons", of which there are 50 bazillion copies laying about) is a boundless font of information.
Otherwise, head for Vivec's Palace.