Hello! First off, let's get the simple stuff out of the way. This is a build that abuses Anivia's ultimate, Glacial Storm. With an extremely short cooldown, it makes this a great area of effect ability to abuse and use everywhere. The only downside to it however is it's very heavy mana usage. Well, that's where this build comes in.

This build is super simple and super easy once you play it a few times. However, when you first begin, you must do things right, otherwise you miss out and can't get the momentum that you'll need. Let's go over some of the things that you have to know.
1: Cooldown Reduction. This of course puts an even smaller cooldown on Glacial Storm, and you can have it down to 3 seconds, which isn't quite spam-able, but as close as you can get it with Anivia!
2: Mana Regeneration. This is incredibly important for being able to stay ontop of things at all times. With the cooldown being so low for all of your abilities, you'll want to be able to afford the Mana that it will cost you to spam all of your spells, and still keep your Glacial Storm up during the whole fight!
3: Mid lane. This build works wonders when you are going mid, and as odd as it may sound, you WANT to be alone as Anivia. This may also sound greedy, but people killing the minions in your lane will put a big halt to this build, as you need as much gold as you can get in order to afford the Mana items to be able to spam Glacial Storm sooner. It's bad enough the minions do it, we don't need players to do the same. Anivia fits best with this build in mid lane because she can hold it off well against 2 enemies, and in some games, i've been able to hold off entire teams with this (hiding behind a turret, of course).

Alright, so the strategy in a game is super simple. In a regular 5v5 game, you start the game with 50 less gold than you need to buy the first item. No problem! Don't buy anything, just go straight to your lane. Now, as long as you are atleast decent with Anivia, you should be able to hold your lane rather well and occasionally pick at your opponent with a well placed Flash Frost. When doing this however, if you can, try and have the Flash Frost run through some lane minions too just for a little extra damage. If their is no enemy in your lane for some odd reason, feel free to waste your Flash Frost to deal a good deal of damage to the lot of them, and perhaps even kill them! It's no big deal in the beginning since you'll be teleporting back and forth to buy items. Later game, however, don't do this, but we'll get to that.
Once you get 550 gold, teleport back and grab a Forbidden Idol, but try to make sure that your lane wont be pushed immediately as you leave. If it does get pushed, no huge deal, just get back there quickly. Stay in the area until you are full health and mana and then head back to your lane. Stay there until you have enough to get your next item, and so on and so forth all the way to the end.
As far as abilities go, just follow the guide above and you should be good to go. One VERY important thing to note however is: Don't be afraid. As Anivia, you have tons of ways to slow and stun your opponent(s). Don't ever run in fear, run with purpose. If you are in a lane and there's 3 enemy champions chasing you, run backwards and use your abilities behind you towards them. Glacial Storm will slow them all, your wall will buy you time, your flash will boost you forward if they get too close, your ignite will do damage over time, your Flash Frost will stun them if they are too close together, and if they get low health, then your Frostbite will finish them off. If they turn to run, turn around and finish them off if you can. And if you've followed the item build correctly and you are towards the end, you have Frost Queen's Claim to slow them down even more if you need it. You are amazing when dealing with their team grouped up, even on your own. Don't ever fear them, just know what you are doing.
One more final thing to note. Once you get level 6 and get your first point into Glacial Storm, use it. Go to the group of enemy minions in your lane, finish them off if they are almost dead, and when that new group comes in, use glacial storm in a spot where it hits all of them, and do this every time you have enough mana, because trust me, this will level you faster than anyone else in your game should be leveling, unless they are uber uber good. I've managed to be level 18 with this build by the time everyone else in the game hit 12. It doesn't take a lot to figure out who won that game. Having a huge level difference like that can almost gaurentee you the win of any game. Again, do not be afraid to use it. By killing all those minions over and over in their group, you get tons of gold, pushing you closer and closer to your end goal: Infinite Glacial Storm. Have fun with this everyone, and don't forget to favorite this build if you like it and maybe leave a comment as to how I could make it better (If possible)!