This is my build for when I'm playing Anivia.
This makes her a late game nuke, and makes sure you won't get the problem of her weaker skills (like health)

This is pretty much the easiest part to get.
It's just heavy ap/level runes.
So that you will be able to start nuking the enemy sooner than with any other kind of runes.

I start out with a sapphire crystal and 2 health potions.
The sapphire crystal helps me buy the tear of goddess sooner, while the health potions make sure I stay alive long enough to get the gold and level.

Then I either buy the boots first or the tear of goddess. (depends on the situation)
You just have to get the tear of goddess soon because you will get in need of mana.

Then catalyst the protector, sorcerer's shoes, rod of ages.
Again I either go for rod of ages or the sorcerer's shoes first. (depends on the situation)
Sorcerer's shoes for her movement speed and the magic penetration.
Rod of ages for the health,mana and ap.

Then I buy archangel's staff, when I've bought archangel's staff I haven't got any problem anymore with mana.
+ you get a reaaallly nice amount of ap.

And then my build is greatly finished... So I start building on rabadon's because of the great
amount of ap.

I max flash frost first, because it is a nuking skill.
You can constantly harass your enemies with that skill.
Along flash frost I go for Frostbite since it is really useful if you're trying to finish of an enemy champion.

I personally prefer getting crystalize later in the game, because it's useful to escape and not letting the enemy escape.. But flash frost and Frostbite are more important.

And of course ultimate skill at 6,11,16

I pick clarity and teleport as my summoner spells.

Clarity is useful early game when you've reeeaaallly got a problem with mana (if you're constantly harassing)

And teleport because of one fact; If you're gone the enemy champion gets to get gold and xp from minions.
With teleport you can recall, buy something, and be back at the same place before the enemy champion even figured out you were gone.

Farming is really easy when playing as Anivia.

Anivia has the ability to constantly harass an enemy with her Flash frost.
Most players will , after Anivia fired flash frost, go back a little; So that you're able to farm a few minions.
You could also use her Ulti to kill minions very fast, this of course has one problem... the ulti eats mana.
And if you haven't got tear of godess or archangel's staff yet and get ganked right after using this technique, you won't have enough mana to escape.

But still the easiest technique is: attack with flash frost to make te enemy champion go back a little and then take a few minions.

This build is aimed at mid players.
But late game you probably will have to get in team fights.
If so, use your ultimate to get everyone of your back.
Then use crystalize to make sure the enemy doesn't escape, if you succeed in doing that use your other skills (especially your ulti) to finish them of.

If you are attacked and nearly dead, look if you got your egg.
If so go to a turret, and die.... The enemy will probably attack you and get more damage from the turret then you'll get from him.
This makes sure once you get alive again, you can finish him off.

This build is aimed at mid players,
though it also works for laners.

I'd say use this build to your advantage, and make freeze some ***.