This is a build I used to go 22/1 in ranked and just recently used to go 26/0 in a normal 5s. I have tried many a build with Anivia and I still find this one to be my best.

Early game you're going to start with a Sapphire Crystal and two health pots. You may ask why you do this instead of a Doran's Ring. My answer for you is you already have enough AP and mana regen to suffice with just the additional mana. Plus your Q when timed right hits twice.

Mid game you should hopefully have your rod, a decently stacked tear, and your boots. Anivia is one of the squishiest champs in the game so rod of ages is one of the most important items in your build to beef her up a tad. The tear gives you the extra mana you need to spam the heck out of your abilities, however, you should still ask your jungler for blue whenever possible.

In the later part of the game you're either fully built or building into your void staff. You're focus during this part of the game is to be an asset in team fights by stunning high priority targets, laying down your ult for the AoE slow and damage, spamming your E for damage, and most importantly walling off exits to prevent escape or to prevent chasing.

As Anivia your main combo will be Q, R, E, W and then basic attacks as they try and run around your wall at which point you can hopefully begin the process over or at least land another R and E into ignite for a kill.

Anivia's passive makes her very hard to kill early game so you should take advantage of this by playing fairly aggressive in the lane. You should farm and land your Q E combo as much as possible. When you get low see if you can't bait your opponent into the turret for an easy kill while you just sit in egg form.

Mid to late game you want to make sure that when you know you're going to die you want to be able to either flash over a wall so its hard for them to get to your egg or flash under your turret so they take heavy damage doing for you.

The last and probably most important mechanics tip I can give you is that she is incredible bursty with her E when the enemy is chilled so try and chill them before you pop the E. This also makes Anivia a good turn around champ. If you're being chased, run into a bush and unload on them before you run again. My advice is run between cooldowns.