Back Line Anivia pretty much does what the title says. Early game, start at mid, farming with last hits.

Before I start, I just want to say that this is not an Anivia skill guide. This is a guide on how to play Anivia as a disruption role.

Try not to poke with your Q until level 2 because you'll want to conserve your mana.
The moment you hit level 2, let your opponent know who's boss. Allow them to believe they can push you back, and when they do, stun them with your Q and land your E for the combined damage. It won't be enough to kill your opponent but it will scare them out of wanting to farm.
If you can land another Q and follow up with your E then try to ignite them. If they haven't healed too much from the first combo then that will be enough to get a quick, clean kill.
If you're level 3 by this point, do be afraid to place a wall behind your opponent in order to land an auto attack, or get some minion/turret damage.
Also, always remember that people forget about Anivia's passive, so baiting them to tower dive, then walling them in is a very easy way to get a kill.

Just keep gradually farming, leveling up your wall and your E. Your E is your main source of damage as Anivia and a big wall early game makes a grand difference in team fights. It can be used to separate champions long enough for your team to get a kill and it can also save your life if it comes to that.

Once you hit level 6, this is where the kills start getting easier. Try not to push your lane too quickly with your ultimate. Rather, bait your opponent to around the middle of the lane then land them in your ultimate. Immediately after placing your ultimate on them, place a wall right behind them, disabling a quick escape (without the use of Flash or other jumping methods) then follow up with your Q and then E. Hitting your enemy with your Q under your ultimate should be much easier since they're slowed down. The overall damage should be enough to take down a champion at full hp, and if not, just Ignite them or auto attack to ensure the kill.

While they re-spawning and returning to lane, this is the chance to farm about 2 or 3 waves with your ultimate.
Use your mana potion now if you haven't already to keep your mana up a little longer until your opponent returns.
By the time they start to return, run back a safe distance and recall. Your minions will be pushed up enough for you to not have to worry about your tower taking much, if any damage.
Buy as much as you can afford, and any items that seem necessary to the situation (Wards, potions, etc.)

Rinse and repeat this tactic until the laning phase is over but constantly be aware of potential ganks and use your wall/stun/ultimate to get away. The better you're doing, the more the enemy team will want to shut you down.

Now, when team fights start, this is where you start becoming a pain for the enemy team. While your damage isn't extraordinary, yet, your ability to turn the tides of a fight to your teams favor is.
At this point, your wall is probably pretty big, right? This is why you've been leveling it in sync with your E.
Upon engaging, wall out any enemy champions you can, potentially cutting the enemies damage by anything from 1, to 4 champions. This will give your team a ridiculous advantage as they get an immediate ensured kill if you do this correctly.
As you can see in the picture, I placed a wall between Annie, Trundle and Cho'Gath. This prevents Annie from using any pesky stuns or her Tibbers, as well as trapping in Trundle for a sure kill. Meanwhile, Cho'Gath takes extra damage from having to run through Anivia's ultimate. Lastly, it gives Nasus enough time to escape Annie's burst, enabling us to escape without any casualties.

Save your stun and other techniques for when your wall melts. The power of your team onto 1 champion will be enough and your abilities will be needed very much to clear the rest of the team.
Start off by landing your ultimate in the battle zone, and land a Q on the most dangerous champion, or if possible, try to stun a champion still attempting to enter the fight.
Use your E on the most dangerous champion if they're under your ultimate.
You will also need to constantly keep a safe distance from the fight and just poke and stun whenever you can. Your walls and ultimate are essential to winning team fights which is why you need to stay alive.

When you have your Tears of the Goddess, start proccing it whenever possible. I usually use my Q, W and R at the spawn and spam my R until I come close to a point where I'll need it.
Your damage starts to become impressive by the time you get your Deathcap which is even more of a reason for you to keep your safe distance from team fights while you have your ultimate up. The attack speed reduction and constant damage are dramatically important in late game teamfights.

One you have built up to your Archangel staff, your ultimate should be averaging around 220+ damage per second with your combo hitting an average of 700 damage.

Try to focus on disabling the enemies more annoying champions during a teamfight, like Fiddlesticks. Generally champions with powerful AOE / DOT skills.

You're probably asking yourself now, "Anivia ganks? How would what work?"
Well, as unlikely as it may seem, at level 6(+), if you're doing well in your lane and have mana to spare (about half a bar at least.) then you can pretty easily get an assist/kill on another lane to help your team out.
Start by alerting your team mate(s) that you are coming in for a gank, and make sure you leave your lane outside of your enemy's field of view. After this, wait in the nearby bush, waiting for an opportunity to strike.
I usually engage with a successful Q, quickly following up with a wall, blocking and or separating the enemy champions. I then place my ultimate and land my E on the target, making for a quick kill.
If you're not too confident with your ability to land your Q, don't engage with it and allow your teammates to engage, then make your appearance and use the strategy above. If you do however try to engage with your Q and miss, then immediately leave the lane. The whole effectiveness of an Anivia gank is the element of surprise with her ridiculous CC. If an enemy is caught they will have no choice but to try and flee, however if you show yourself then you make yourself an easy target to the enemy if you don't flee immediately and this could potentially cripple your allies rather than assisting them. Without your Q, you'll have to resort to your ultimate and your wall to escape which is a severe waste of mana and time for an early game escape.

It's somewhat rare that you may want to try an Anivia gank, but they can easily be performed if you have your team's jungler show presence in your lane. Portraying the illusion that you have recalled to buy items, etc, leaving the enemy off guard to ganks.

There isn't really very much to say about this that anyone who's played Anivia before wouldn't already know, but for those who are still unaware of how Anivia is an excellent escape artist for herself and her team, these are some tips I can give.

1. Anivia's wall isn't only great for trapping enemy champions from escaping certain death. It's also a brilliant tool for trapping enemy champions in a linear path (i.e. Jungle, spawn exits). If you or your team is on the verge of dying, don't be afraid to wall off the fight and give your team a brief opportunity to escape and recuperate. Sometimes it's best not to risk losing any teammates, especially if they have a buff.

2. A well fired Q can save you or your allies from oncoming enemy champions. The plus side of this is that if you hit it, you can very easily get away from whomever you stunned, saving you and or your allies. The downside is that you momentarily stop moving in order to fire your Q, and it's not ensured to hit the enemy champion, so if you miss, you're almost definitely going to die. I wouldn't recommend this escape tactic unless you can get away unscathed even if you do miss, or if you're positive you can hit the enemy champion(s).

3. Anivia's ultimate is a great way to stop enemy champions from chasing you and your allies down, especially if you place it in a position that the enemy has to run through in order to get to your or your team. Just like your Q, you pause momentarily to place your ultimate, but the time it buys you is definitely worth it. Try to place your ultimate in front of the enemy, rather then on them if you're escaping. Even if they don't run through it, they will still run around it, wasting time, and giving your team the time they need to escape.

Honestly, as Anivia, my favorite spell is her wall. I find it to be SO useful in almost all situations, especially when it comes to engaging. When I'm on Skype with my friends, and I'm playing as Anivia, I will always try to engage the teamfights with my wall by separating any split champions on the enemy team.
I spoke about this earlier, but sometimes walls won't go as perfectly as you planned. If you're not careful, you could separate members of your team and give the enemy the advantage you were trying to give to your team.
Try to use your wall sparingly. Make sure you know that you're going to separate the right champions and not ruin your own team's chances of success.

One of Anivia's biggest concerns when fighting is her mana. If playing Anivia as a back line role, then you don't need to worry to much about your hp and you shouldn't really be finding yourself in too many situations where you're going to take allot of damage.

Anivia's damage is already very potent with her spell combo but that alone doesn't compensate for your opponents variables (i.e. runes, items, champion abilities, etc.). That's why building magic penetration and ability power runes gives you that equal advantage early game to compensate for the lack of AP items you have early game.
Early game Anivia you should really just focus on quick bursts and farming rather than blitzing out your mana every time you see an enemy champion.

Starting with a Meki Pendant and 2 potions gives you enough mana regen to poke your enemy enough to negate them and or allow you to farm. The health potion is just there to help if you need that little boost to get a killing blow on an enemy champion. You may ask "why would you need a health potion if you have your egg?"
Well, your egg is a handy little way of escaping death early game due to the strength of your opponent, especially after bursting you into your egg, but it's much safer to save your egg for a time when it's critically needed. That's why spending an extra 35 gold on a health potion seems to be the most lucrative way to spend your money at the beginning.

By the time you start building AP, you will have a ridiculous amount of mana regen. So much so that in some scenarios you can donate your rights for blue buff to another champion who could make better use out of it.

I've already explained the purpose of Ignite earlier in this guide, but I forgot to mention it's effectiveness when used on an enemy inside your ultimate. Anivia's ultimate + Ignite quite figuratively burns through the enemy champions hp faster than they have time to react too. This combined with the power of your stun + E combo, it's rare that a champion other than a tank will get away from you.

The reason I've chose Flash as a spell with Anivia is to compensate for her lack of speed and to make an easy get away.
Often, when I can, I'll place a wall in front of me and then flash over it if I desperately need to escape from certain death. Most enemy champions are momentarily stifled by this and end up giving up the chase or burning their own flash to get to you. At which point you can just stun the remaining pursuers and safely escape.
I also find combining Flash with one of Anivia's many CC spells to be very effective. For instance:
Anivia and ally chasing down enemy who escaped from a fight. Their escape seems inevitable until Anivia flashes closer towards the enemy, firing her Q at the same time and stunning the enemy champ. Anivia then uses her ultimate and wall to keep the enemy in place long enough to land her E and allow her ally to get the kill or assist.

Blitzcrank - I love playing Anivia with a Blitzcrank. If Blitz can land a successful pull on an enemy champion, I can wall them in preventing any possible chance of escaping. That, combined with Blitz's knock up makes the enemy champion an easy target for my stun and combo. Enemy champs very rarely survive this combo.

Nautilus - Pretty much the same aspect as Blitzcrank. Combine Nautilus' pull and stun with my wall and stun makes for an easy easy kill early and late game.

Darius - Another character with a pull, but it's not just the pull that I appreciate from Darius. It's his passive that I find works best with Anivia. His constant DOT + my ultimate's DOT makes for a pretty quick kill. It's also easier for Darius to land his passive and abilities while an enemy champion has been slowed/stunned/trapped by me.

Generally any champion with a good burst works brilliant with this Anivia build. This build is all about CC so the quicker the enemy champion can be killed, the better.

Pros:

You can save your teammates lives very easily.
Your presence in team fights makes a massive difference.
Enemies will try their best to get to you, making themselves easy targets for your allies.
You can negate an enemy champion from farming early game.
You can ensure kills for your team by trapping your enemies with all your CC.
You can clear waves very efficiently, making ample distractions for your enemy.
It's not too difficult to fight in team fights as you can keep your composure from outside.
A well farmed Anivia can do very good damage to an AD carry or support early and late game.
A single wall can change the tide of a teamfight.

Cons:

This role is very team reliant.
You are pretty squishy and can be taken out pretty quickly.
A single wall can change the tide of a teamfight.
Without mana, you're almost useless.
You don't have much speed.
You're prone to ambushes.

Q: Why do you not level your Q as much as your other abilities?

A: I find the main use of Anivia's Q is it's stun and ability to proc her E. It's damage isn't nearly as beneficial as the length of an increased wall or the strength of Anivia's E.

Q: What should I do if the enemy's jungler starts to babysit my lane?

A: Ask for the support of a teammate, preferably your jungler if you have one. Have them wait in one of the 2 bushes for an opportune time to strike. You can make this chance yourself by baiting an enemy champion close to your hidden teammate and then separating the 2 champions with a wall, stunning your target and allowing you to pick up a quick and easy kill. Even if you don't get the kill, you can at least scare away the enemy and get back to farming.

Q: The enemy champion is shutting me down and I can't land a stun. What should I do?

A: Just farm with your ultimate. If you can't kill, or negate the enemy champion's farm, then don't feed him either. Just continuously farm with your ultimate and stay at an even pace.

Q: The enemy team is focusing me in team fights and killing me too fast, how can I counter this?

A: Instead of building an Athene's Unholy Grail, just delay that and replace Rylai's Crystal Scepter with something like a Warmog's or Zhonya's Hourglass. Either works brilliantly as you can pretty easily proc your Warmog's with your ultimate, or you can place your ultimate in a teamfight and remain invulnerable while doing so. You won't be able to use any of your other abilities, but your ultimate is still doing it's DOT and slowing the enemy's attack speed and movement speed ensuring a greater chance for victory.