I have been playing anivia for a while :) I also died only a few times. I had a lot of clean death mathes with many kills :) I hope you all like this build.
This is also my first guide so. :P Tips are welcome!

For runes I take some Magic penetration/Ability power and cooldown reduction.
I get magic pen to deal a huge amount of damage.
They Cooldown reduction runes are needed to use your skills faster. I noticed that without it looks forever till you can use your Q again.
Ap per level just to have some extra AP

The mysteries are focused to deal lots of magic damage!

As you can see I focus to get Catalyst the protector first and then I get some movement speed 2 boots. I do this to use the passive of this crystal :).
After I get that I get the sorcery boots that gives extra Magic penetration!:D
Then I go for Rod of ages and Rabadons Deathcap for more pain <3
As you noticed I also get Guardian Angel. Why would I need that? Anivia already has its egg.
Well Guardian Angel Gives you some extra Armour/Magic resist it also revives you again! <3
With Guardian Angel I had survived many times. You resurrect and bam Q-E Combination or Wall/Q and fly away! :D
After Guardian angel I get Will of the acient for some spell vamp it really helps you. If you see a minion wave you just pop up your ult and heal your self quickly :)
And for my last item i get void staff for extra penetration that means more damage! :)

For the first skill I go for Q it gives you a long ranged attack(so you can poke who ever you are fighting with) and then I focus maxing E first :) but I get wall at level 4 It is really handy too use it for your view distance or to block your enemy.

Q is the skill you mostly uses to stun and poke from a long distance.
E is the skill you uses when someone is frozen to hurt the enemy badly!
W is the skill that you can use too save your team mates/Yourself or to block enemys who want to get away! It is also handy so expend your view :P
R is the skill too farm and deal some dmg/slow your opponent!

The spell combinations I mostly use are.
Q-E-R-W (Stun/dmg,Lots of Dmg,Slow,block) and repeat if possible :)
Q-E Just too hurt someone in my lane!
Q-W Too get away from Champions chasing me :S Q to stun and W to block

It is easy to use but hard to master. For Q you need too be able to guess the direction your opponent will head too. I played against a AP yi ones its a good example :) Yi is fast and has a annoying Q skill but ofcourse if you watch carefully you use your Q where Yi started his Q skill and stun him / deal lots of dmg so he only Poked you and you almost killed him!:)
Enemy champoins won't stay and wait for your Q too hit make sure you are able to guess the directions!

I use Flash and Clarity.
Clarity
Most of my team mates say its bad to have Clarity as anivia but I say its perfect.
Clarity gives you mana and Anivia needs lots of mana. It will let you survive early game a lot and even late game if your team mates won't give you blue buff! :S
Clarity can also save your team mates :O!

Flash
I use it to get away or too use Flash-Flash frost-Flash bite combination when a enemy tries to get away :P

Well this is how I build anivia and get away from dieing :)