How do I do that? I completed a quest, where I became archmagister by killing the previous one, but the quests still sits there, even after cashing it and having the archmagister title in the faction menu of the inventory.

You need to speak to Aryon about "Magister" after officially being named Archmagister to close the quest.

If you're talking about replacing Arch-Mage Trebonius, the quest will close only if you duel him in the arena. Otherwise, the quest remains open.

This is the problem, I spoke to Aryon, I've been named Hortator and I'm halfway through killing the dagoths for Vivec...

Specifically, asked about "Magister" after you were officially granted the title.

What has a Mage's Guild to do with the Telvanni? I don't understand how.

http://en.uesp.net/wiki/Morrowind:Arch-Mage is the quest the OP wants to remove from the journal.