So I have been playing Anivia for a while now. My elo isn't high at all but my team competitive 5's is on a very professional level. For this build I want to suggest a build that gives stability and damage. Anivia is a very unique champion who is arguably one of the strongest pushers in the game. Hope you all enjoy!

Pros:
Strong Pusher
High Ranged Abilities
Strong Single Target burst
Strong AoE Burst
Lockout Abilities
Good CC
Cons:
Mana Starved
Countered by Mobility
Long Cooldowns

Anivia is really mana-starved and in this build, it's all about having Mana Regen while keeping solid damage. One of the strong points of Anivia is her damage ratio's and the power she has with her E. The reason I choose these runes is because the early AP+Penetration allows you to chunk the enemy mid for half his/her hp with Q-E and still being able to use abilities to farm thanks to your Mana-Regen runes. Anivia excels at zoning, so these runes really utilize her best.

Alright this is where it comes down to in the end. The Build
Usually you always start Boots+3 Health Potions
If you are losing or going equal in your lane you want to get Chalice of Harmony
If you are winning your lane you should get Blasting Wand
I'll explain this. If you are losing or going equal getting a Chalice will allow you to endure his harass and grab more CS thanks to the Mana Regen+Blue Buff.
If you are winning then you can rely on blue buff for mana and use Blasting Wand+Blue Buff to push back the enemy and thereforth get ahead.
If you are still losing or going equal in lane, you are going to grab Catalyst the Protector
If you are still winning you'll want Fiendish Codex.
For the reason of Catalyst it's the same as Chalic.
Grabbing Fiendish Codex will allow you to spam abilities more and have a stable Mana Regen output which will amplify the amount of harass you can use.
The next item you will get is Athene's Holy Grail. This item gives you MR, AP, MRegen, and overall everything you need. After that It's wise to pickup Rod of Ages because this will give you HP without needing a Warmogs and will still give you Mana+Ability Power.
After this your going to get WoTA(Will of the Ancients) this item is so good for anivia as it gives her a ton of sustain especially with her high burst capability.
5th item should either be:
Rylais Crystal Scepter: If you're winning and ahead
Abyscell Scepter: If you're getting bursted hard in team fights
Rabadon's Deathcap: If you're snowballing really strong and aren't getting focused at all
6th Item should be:
Rabadon's Deathcap: If you didn't pick it up 5th
Rylais Crystal Scepter: If you picked up Rabadon's 5th.

When it comes to summoner spells, a lot of spells can work for Anivia but none like the 2 shown above.
Flash - Good on every champion, especially good for anivia as she can Flash-Wall and then nuke an enemy mid. Also great considering she has no escape mechanism other than her wall.
Ignite - Amplifys her damage and is great for securing skills. Almost every mid gets this to add to their burst capability.
Exhaust - Not a bad idea on Anivia and certainly can utilize well but will not be too useful considering they can always flash out of your ult meaning your exhaust will more-or-less go to waste.
Heal - Not bad for baiting but Anivia already has her passive for that. All preference though
Ghost - Anivia doesn't benefit on the MS because of her long cooldowns. Once she bursts she's gone so she has no need to have a skill for chasing when she has Stuns, Slows, and Walls.
Surge - Not bad on many AP carries but just doesn't give the DPS that you can get with Ignite. Anivia doesn't benefit from the AS either.
Clarity - Not bad for Anivia but with the items and runes you should have plenty-fine mana with you. Enough to play smart and not need to waste a summoner on Clarity.

Your technique is usually going to be E-R-W to bait into Q
Remember if you toss your E and then R immediately when the E hits it will get Double Damage because the R apply's slows instantly.

Your going to want to max E as it's your main output of damage. Then you're going to want wall to single out targets in mid. It can cover almost all of mid lane so remember that or you can max Q if you're confident in your ability to land a stun.

R>E>W=Q is your Skill choice order.
E-R-W-Q is your first Combo
Q-E-W-R is your second combo
Q-R-E-W is your third combo

The first combo should be used for harass
The second combo should be used for 1v1 Fight off's
The third should be used in team-fights.

In Summary Anivia is a high burst champion with huge push pressure and kill potential. She is a Top Tier Mid Choice in any Elo and competitive play. I hope you all enjoyed my build and I'll see you on the fields Summoners! -EvilBeanDevil