So, as most Anivia players probably know...Anivia is a nuker, a ganker and an overall hated foe. It is entirely possible to not only carry your team but also fend off multiple opponents at once with a proper build. While this build is probably NOT the best, i assure you that if properly executed it can leave a wake of destruction.

Starting off with runes. Runes are not the make or break for this build, but every little bit helps so i'd advise the following. 9 greater marks of insight for early magic pen. 9 greater seals AND glyphs of potency for early game AP. Finally 3 greater quintessences of potency. Now im sure there are least a few Anivia players that are screaming "WHAT ARE YOU DOING!? USE THE QUINT OF FORTITUDE NYEAHH!!!" But you can just ignore them. While that IS a way to play Anivia the best way to get a decent build is to get some early EARLY game kills. (30 Seconds after minions start to fight in a 5v5 or sooner, pre-minion spawning in a 3v3)

I like to stack runes that give flat bonuses. No one cares about an extra 10 AP late game when you have 500. If you buy an amplifying tome at start with this build u can boast a near 50 AP starting game. In combonation with techniques i am yet to present that can be devastating.

Masteries are important for Anivia because of two main points. 15% Magic Pen and everything in the Utility tree. IMO The utility tree is by far the best for almost all casters and there are a lot of perks that come along with using it(Such as the extra tiny bit of XP, movement speed etc.

These masteries take advantage of Anivia's perks to the fullest

Item Building is a make or break for Anivia. I usually hit items in the following.

Amplifying Tome (Early AP)
Sorcerer's Shoe's (Magic Pen and Speed)
Rod of Ages (AP Mana and Health)
Archangel's Staff (Mana Pool building and AP to follow)
Rabadon's Deathcap (Insane AP boost and game turner)
Lich Bane (Movement Speed increase and Attack buff)

Now you may be wondering "Why does this noob have 7 items?"
It's cause i sell The boots after i get Lich Bane.

Then buy Hextech Gunblade.
If you ever get this far into your build it is GAME SET MATCH.
Be sure to take advantage of the increased movement speed slowing and sapping of your opponents. Ive only gotten to do this build a few times and it usually comes alongside the LEGENDARY sound-effect.

Skill sequence is super important. Be sure to get Flash frost first to stun. Then Frostbite for Damage (2x when frozen) and then Crystallize (The Wall). Dont focus too much on the wall once you have it, just leave it for last and get the damage dealing moves up first. If you are having a hard time using it effectively feel free to build out of order and lvl it up once or twice.

Glacial storm at lvl 6. And again whenever possible. This is by far the best move Anivia has for CC and ganks in combonation with the wall. In late game (with maxed skills for everything) i recommend the following combo.
Glacial storm an opponent placing it where they must travel through it to escape your teammates. Floow it up with Crystalize and Frostbite. Then Stun with Flash Frost and Frostbite again if you have the time. If all five of these skills are executed it is almost certain that 1-3 opponents are now KO'd and they are probably reaaaaallly hating birds.

I like clarity and heal just because they let you stay out early game for much longer. Other than that there isnt much depth.

While it is certainly possible to Jungle, i highly discourage it. She is far better used to gank and farm early game. Get blue buff whenever you can and use it to spam abilitys.

Anivia's Glacial storm is GREAT for farming and trapping people. Use it to gank heavy in 5v5 and protect towers in 3v3. She is a skillshot hero, which makes her one of the hardest to play, but she also has some of the best potential. Get used to the timing and everything else will follow.

Overall Anivia is my favorite Hero to play. She is OP when she is fed and i love getting into that situation. When Jax and Garen flee from you at the same time when you have half health...pat yourself on the back. Anivia has been mastered.