Anivia is one of the highest AP damage champions in the game, this build takes advantage of that, supplementing it with high magic penetration and a moderate dependance on Soul Steal Stacks. This build assumes your playing a 5v5 and laning with a partner.

I start each game with a Sapphire Crystal one Health potion and one Mana Potion taking them as need until I've built Rod of Ages, this helps keep Anivia alive and relevant in lane, also taking Clarity for the same reason. Flash is a useful spell for both offense and defensive purposes and will likely get you your first few kills and potentially saving you from ganks, Crystalize is also a good anti gank tool, learn to kite with your stun and Crystalize.

Unless you have a good teammate you won't be getting a lot of kills early, game but once you hit level 6 and build Catalyst with Sorcerers Shoes you'll be able to deal significant burst damage using Glacial Storm / Flash Frost in conjunction with Frost Bite.

Once you have Rod of Ages and Soul Stealer built its time to work on your Magic Penetration, Abyssal Scepter and Void Staff complement your Magic Penetration Marks and Sorcerers Shoes well and will allow you to severely damage otherwise magic resistant or high health enemies such as tanks. Note that you won't be able to solo them but taking out a quarter of a tanks health is a great benefit in a team fight.

***If at this point you haven't gotten a lot of stacks don't worry, your damage should be relevant.***

Late game you should have one remaining item slot, this should be filled with a Needlessly Large Rod or a Blasting Wand, the later preferred and if the game last long enough finish off the build with a Rabbodon's Deathcap.

Helpful Hints

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Anivia requires good map awareness to take full advantage of her burst damage early game and avoid being ganked, you'll want to hop lanes and assist allies especially when opponents over commit to a turret. Be sure that your lane partner can handle the lane in your absence.

If your opponent chases an ally down a lane or through the jungle aiming Flash Frost at them while heading toward the pursued ally can potentially save them and sometimes get you a kill or an assist if you stun properly.

Anivia requires fast reflexes and timing, try laying down Glacial storm and blocking opponents in your AoE with Crystalize. Forcing them to stay in your AoE is a great way to get kills or force enemies away from a kill. Many times you can force a foe to walk in an undesirable direction as they auto walk to get to their prior destination, take advantage of this.

If you're in a lane with opponents hugging their turret you can use this to your advantage by laying down your AoE on their turret and hitting them with Frost bite, this will deal a good amount of damage and often times cause them to back off of their turret.

If your having trouble getting kills for stacks, assists count for two stacks, stay behind and ally and attempt to help them kill an opponent.

The blue buff is extremely helpful throughout the game but far more so early on.

Anivia is a good Solo lane champion for mid or a jungler however you'll take some losses in AP to sustain her high mana cost, for this I rush a Tear of the Goddess first and then continue this build as normal, building an ArchAngels Staff after I have my Soul Stealer, but you won't have a Deathcap to back up your soul stealer, should you build this way. You'll also be more prone to attack with the delay in attaining Catalyst.