Anivia is a champion which depends more on player's skills. Riot's setting her difficulty as 10 doesn't mean that she can't be use well by any player. Mostly you can think of Anivia counter champions but she doesn't really have a counter. Using her correctly makes her uncounterable by any champions.

Pros

• Has high enough health to survive team fights(if cast skills right)
  
• Higher Damage output
  
• High Mana for long team fights
  
• Can Fight Solo Versus Many enemies

Cons

• Mana Greed at Starting game
  
• Tactical Build
  
• Requires High Strategical Skill from players
  
• Requires Stable Game Connection
  
• Requires Focus

Expose Weakness amplifies team damage by using Glacial Storm and Flash Frost.
Oppressor amplifies Frostbite burst damage when enemies are slowed or stunned.
Precision gives additional magic penetration.
Thunderlord's Decree burst damage every 3rd skill or hit. Also gives additional damage to champions or build that doesn't have high enough ability power or attack damage . Otherwise use Deathfire Touch .

It depends on your play style but still this build requires high enough magic penetration to remove the damage reduction provided by Glyph of Magic Resist or Glyph of Scaling Magic Resist.

For Aggressive plays
Flash and Ignite sure can take the kill.

For Team plays and escapes
Teleport and Flash can be used to roam around.

Rushing to build Tear of the Goddess will help you to secure you have ample of mana of mid to lage game team fights.

The do not build Archangel's Staff immediately. Tear of the Goddess still stacks mana. buying it as early as 12-15 minutes doesn't really do anything because Tear of the Goddess still has 200-400 stacks of mana.

Instead buy a Rod of Ages it fills the time for the Tear of the Goddess to stack nearly full.

Buy Boots and Haunting Guise if Tear of the Goddess's stack is at 500-600.

Haunting Guise gives you magic penetration and health you will need in fights.

you can also buy Chalice of Harmony instead of Haunting Guise if your enemy has high damage than you can take in early game.

Buy Abyssal Mask if you have enemies like LeBlanc who needs to go near you for her burst. it gives magic resist and also additional magic penetration to burst them before they can burst you.

Banshee's Veil and Zhonya's Hourglass are helpful if enemies has AoE Crowd Control or High Damage burst like you.

Start with Flash Frost to avoid early death. Don't waste Rebirth for nothing at an early game. Rebirth reduces Anivia's armor and magic resist at early levels.

Maxing Frostbite ensures that you have burst. Frostbite doubles it's damage if enemies were slowed by Glacial Storm or Flash Frost.

Anivia is prone to ganging. she can only burst one at a time. so always coordinate with your team to avoid surprise attacks from the enemies or for your team to help you escape the bitter fate of getting slayed.

Try to land Flash Frost and Glacial Storm with multiple enemies. Crowd Control from these two skill will help your team win in team fights.

If your jungler offers you blue accept it. even if Anivia doesn't need blue buffs in this build still it's rude to decline what is gifted.

try to position Crystallize to the best place as possible. Crystallize can be use to knock back, wall or wards. use it efficiently.

If enemies were able to get out of your Glacial Storm don't waste trying to spam all skills. you can also trick them by shooting Frostbite and land glacial frost just before the Frostbite hits the enemy.

Glacial Storm increases it's damage if it reaches it's full size. try to hold as many enemies as possible and as long as they can using Flash Frost and Crystallize for more damage.

Releasing Frostbite as ealy as possible allows you to cast another Frostbite just before the enemies escape.

Anivia is a champion who focuses on the user's strategy. Try to use every skill as efficient as possible for more favorable outcome.

Anivia is derived from the Latin words niveus / nivea / niveum, all which mean 'snow white'.

Every Champion Has it's strengths and weaknesses.
On how you use your champion makes them strong.
On how you exploit their weaknesses makes them weaker.