The purchase of the chalice of Harmony serves two purposes, i gives you early game MR (since you typically go up against an AP champ in mid) and it gives you the mana necessary to keep the mid lane pushed up to turret. Keeping the lane pushed up to mid turret does several things

It forces the enemy champ to either use spells to farm or risk losing creeps to the turret. This is important bc it prevents them from being able to effectively harass you. With your E and Q, you can still safely harass when they try to get creep kills.
It allows you to steal enemy jungles wraiths using Q+E
It gives you plenty of time to port your slow *** back to town, buy, and get back to your lane before you get pushed too hard.
When you get your Tear of the Goddess/Manamune, it allows you to fire off a q every 3 seconds to max your mana as soon as possible.

It is important to note that under this strategy, you need to save your ulti as an escape mechanic. when the jungle comes for you (and eventually he will) ulti him to swap positions, throw up your shield, q him to slow him and book it back to your turret. If need be, you still have flash and exhaust. Remember, if he comes to lane to gank and you can get away without using summoner spells, the gank was a fail because he did not grant any advantage to his mid laner. While you will eventually sell the chalice, it gives you the tools early to mid/late game to harass relatively consistently and safely.

The Brutalizer - Once you purchase a brutalizer, you should have around 40 armor pen which should help keep your damage up. More importantly, you have the CD necessary to squeeze off a forth Q when you hit with your E.

Merc Treads - Pure and simple, you need boots especially if you are going to be hugging enemy tower. If the opposing team is really light on AP and/or CC, you may want to consider boots of swiftness or Ninja Tabi instead, but almost 100% of the time you will be getting the merc treads to help counter the enemy AP burst. Combined with your runes and your chalice of harmony, you should be around 99 MR when you hit lvl 6 which should give the the resistance you need if the AP carry decides to try and burst you. Remember that with your MR and your passice, you should be reducing incoming damage from an AP carry by around 51% if they are running MR Pen runes and %58 if they aren't. Combine this with exhaust and their damage is reduced by around 72%. the extra armor granted by your ulti brings this reduction to around 80%. So if they dont kill you with the burst and they are in a good position, calmly engage them. If they are a champ with short CD, pop your exhaust, your ulti, and your W right before their powers are up.

Manamune - Manamune is your next item and will be a large source of damage for you. You want to get the manamune as soon as possible so that you can begin increasing your Mana from the tear. Thanks to your chalice, you should be able to spam q every 3 seconds and never run out of mana.

Sheen is a no brainer. While your Q doesn't apply on hit effects, you are using your Q enough that it will greatly increase your damage output from your auto attack.

At this point your core build is complete. Your next items should be built based on the enemy team. Prior to selling your chalice, I typically like to buy my Glacial shroud next for the CDR and because I already have plenty of MR from my chalice bu no real source of Armor yet.

DO NOT SELL YOUR CHALICE UNTIL YOU HAVE MAXED YOUR MANAMUNE AND HAVE ENOUGH TO BUY A BANSHEES VEIL. Selling your chalice to early will leave your MR too low and eliminates your primary MP5.