This build was something that I fell in love with after the release of the unholy chalice. I played several games and was constantly at 75% or higher mana and I used my ult almost non-stop. The greatest problem to Anivia is mana regulation. With this, it is virtually eliminated. Early on you want to get blue when you can but eventually you can regen mana over 25 mana a second. That's over 125/tick according to your in game tool tip. Add to that the 12% max mana regen per assist and kill... you see where this is going.

The main strat is just to keep your ult down on baddies as much as possible. Use your wall to keep them trapped in it as long as possbile and use that stun when you can. Hit em with Frostbite when it's up and you are dealing mad damage.

Pros:
Lots of mana to play with.
Decent damage.
Spell Vamp.

Cons:
Squishy.
No area of intense focus.

Overall:
The overall is that even though we sacrifice a little bit of AP and survivability she can go for a long time without having to fight for blues late game. Meaning that if you do end up dying you aren't handing your enemy something very useful.

I went with the summoner spells of TP and Flash so you can stay out in lane as much as possible but flash is such a great tool that I almost always incorporate it into every build. Anivia is a slow *** bird so TP will give you as much xp time as you can which allows you to go back for your items as the money rolls in so you can keep your items stacking. Especially for the Tear.

Tear is a rush job. You want the stacks as fast as you can get them. Remember that you get a charge for manually turning off her ult as well. So wait the 3 seconds after casting to turn it off.

The second thing you want to rush is the unholy chalice. That is the main source of your mana regen. But I put boots before because you don't want to be the slowest thing on the map. Keep the reg boots until you can get the unholy chalice. I promise it will be worth it.

The rest of the items should make sense, I hope. In the end if you really need to get that extra punch in or the enemy doesn't seem to focus you much and you feel skilled enough you can take off the Banshee's Veil and go with the Void Staff. Also if there team is more AD focused and they don't have and spells or abilities you are really afraid of then, I would go with Zhonya's Hourglass. The extra AP is pretty awesome.

I put the skills in this order mainly for the increased single target dps. Frostbite deals mad damage to the target lowest which works great for solo lanes and also for knocking that enemy down before he can get away. You want at least 1 level of wall because it is very useful. You can cast it while running and it doesn't slow you down. Utilize this if you need to bail. It's great if you can get the trapped somewhere as well, like against creeps or next to a tower.

I chose these runes mainly because you want to maximize straight AP with no curves for that added early game punch, MPen because well same thing but it helps more late game and then the mana helps keep you regening more over with your kills and assists, and makes it so you can continuously fire shots at the baddies.

The best way to play Anivia is to work around your team. She is just as happy with an assist as the kill. So, use you team to distract them and support them with your AoE slow and stuns while kicking in that heavy single target burst.

The real power of Anivia is she is heavy damage while being a bit of support. Use that. She is also good at kiting. If you have the skills you can slow them, block them and stun them to death why they keep chasing you cause you are just so close.

As with a lot of AP heroes, you have to be careful, because she is a glass cannon. Don't be te lead in fights unless you are confident that you can get out of there before they kill you or be out of reach when your passive activates.

Once you get your unholy chalice and archangel's staff use your ult on everything. It has such a short cooldown and you have a lot of mana regen. Remember, it will help to build those stacks on your archangel's staff. You want to get the full 1000 ASAP. You can take out waves pretty efficiently this way. Also good for getting blue, because your frostbite does double damage to chilled targets, which happens by being in the AoE.