It took a while to make a deck with Millennium Eyes that was competitively viable, but with careful card consideration, this build should be the most reliable Millennium-based deck that you will find. The primary reason for this is that it completely skips the idea of getting relinquished itself ritual summoned, as ritual spells tend to do little more than unnecessarily clog the deck.
With this deck, you rely on searches, draws, and sending the fusion pieces to grave and careful card play to stun your opponent on the second turn. With Dragonic Magic in the deck, it gets a nice buff for the first turn if you have Illusionist Magician in grave. As you can see, the card advantage tends to be very good since you can use Allure, Pot of Desires after sending some monsters to grave, and use Dark World Dealings with little downside (since your fusion spell fusions from the grave). If played carefully, you can get Thousand Restrict on the board as well, which can result in an almost total monster lock.
Play the cards carefully, and try to maintain 1 copy of each of Ill. Magician, Illusionist, and Relinquished in the grave or hand, if possible. Your opponent may try to bait you by throwing out an effect he/she doesn't need, so be wary of this.