This is how I personally build Anivia and I don't expect it to be good. I guess I just feel like posting this up soo... um yeah. :)
Please do not hate if you do not like it or think I am copying.

Pros
-Very nice stun and burst damage with a Q E combo.
-Slows and stuns are really good for escape if landed.
-Liandry's plus her ult is extremely powerful.
-Wall can force enemies around minions.etc and save teammates.
-Late game farm is incredible.

Cons
-Very Mana Hungry
-Bad Mobility
-Dodge Abilities counter her
-Can be hard to land stuns
-Bad early game farm

For runes:
I take Greater Mark of Magic Penetration for the really nice magic pen.
I take Greater Seal of Scaling Mana Regeneration for the mana regen which is useful both early and late and keeps me in lane for a really long time if not harassed.
I take Greater Glyph of Scaling Ability Power for the great boost of AP late game.
I take Greater Quintessence of Movement Speed for better mobility as Anivia has bad mobility.
Greater Quintessence of Ability Power is the main quint for all apc mids.

For masteries, I take Spellsword due to Anivia being AP.
You can also spend the point of Spellsword on Havoc or maybe Destruction .

Boots is needed for Anivia for the speed boost and is the main start for most champions with 3 Health Potion.
Crystalline Flask is if you think you will be getting harassed.
Catalyst the Protector is for the sustain and Tear of the Goddess is for the extra maximum mana + upgradeable late game for a lot of AP and defense. Chalice of Harmony is for if you are having mana problems, though I recommend Tear of the Goddess more than Chalice of Harmony.
Upgrade Boots fast into Sorcerer's Shoes, then build Rod of Ages and Rabbadon's Deathcap.
Buy Zhonya's Hourglass for defense and then upgrade your Chalice/Tear.
Then either buy Void Staff or Liandry's Torment depending on the enemies magic resist.

Get 1 point in Flash Frost for the stun, as stun/cc is most useful early game, Crystallize is useless early, mainly before you get your ult, and Frostbite is far too weak without the enemy being chilled first. Take a point in Frostbite and max it ASAP. I recommend getting a point in Crystallize at level 4, in case of ganks.. just a safety precaution. Take a point in Glacial Storm whenever you can, and max Flash Frost, and lastly max out Crystallize. If needed, take a few points in Flash Frost or Crystallize, whatever you need.

Flash is a main spell for about any champion to get out of sticky situations.. especially with a low mobility champion like Anivia.
Ignite is really good for finishing off enemies, and works well with Glacial Storm.
You may also want to try Clarity because Anivia is a very mana hungry champion.