This is a general guide on the AP heroes that I am confident enough to play competitively with. They are organized into to categories for clarity however there are champions that cross over and can be considered as both burst and utility.

Generally my playstyle is very passive with AP champions. I prefer to out-farm and out-level my opponent and get better items before he does. Usually this kind of playstyle is geared towards mid to late game. Ofcourse there are exceptions to this and the items I get compliment each champion's differences.

Early game:

Brand
Zilean

These champions focus on early game lane and jungle domination. Getting 2 - 3 Doran's Rings here work very well. After the rings I go straight into Deathcap to maintain the early game edge. Playing aggressively is very important to maximize these champions' potential. However if your opponent in lane is out-harassing you or you are in threat of being ganked then back off a bit but maintain the pressure.

Brand has one of the strongest early game phases in the game. Your burst potential at level 3 is incredible, which is why I opt for Quintessence of Potency here. Starting with boots gives you great zone control in lane which is important for aggressive play. While he can pressure his opponent immensely, he is also extremely vulnerable to jungle ganks, but do not underestimate his damage. Many times I have killed the jungler when he tried to gank because he underestimated Brand's damage at level 3 - 5

The extreme case here is Zilean. His burst at early levels is tremendous and he out ranges most laning opponents. He is great at harassing the opponent for free but he falls off hard late-game. The difficulty with Zilean is knowing when to adapt his role into a more supportive type late game when opponents start to stack Magic Resist. Again boots with Doran's Ring allows free harass and zone control. Zilean's range is incredible but if you use up all your mana, your control will diminish. Mana regen yellows helps this weakness a bit but refrain from using his Q -> W -> Q combo if it means running out of mana afterwards and your opponent does not die. Use your Q once everytime your opponent is in range and they will rage. Zilean is quite safe from ganks after level 4 when he gets Time Warp (E). Remember it is more effective sometimes to use your E to slow down your opponent instead of speeding up yourself or your teammate.

Early to late mid game:

Annie
Malzahar

These champions start off weak at early levels but quickly scales up because they have high base damages on their skills. WIth these champions I take ap/lvl yellows and blues because level 9 - 16 is where they truly shine. All these champions take movement speed runes to help in harassing and last hitting without getting denied.

Annie is the perfect example of a great early to late-mid game champion. Once Annie reaches level 6, she becomes VERY scary. Start off the game playing passively and last hitting with Q. Whenever your stun is up and you get the chance to Q -> W without taking too much creep damage you should do it. After level 6 however you can start playing aggressively. A trick I often use is the closer I get to getting my stun stack, the closer I move towards the opponent, slowly building pressure. Once I get stun I am either in range to hit it or the opponent gets too scared. It is important for Annie to farm if she does not get any early game kills. If you do not keep up with farm and get those items, she will fall off hard late game

Again Malzahar is a similar to Annie in that once he hits level 6 he becomes super scurry. Malzahar is extremely strong at jungle invasion. If you get the chance tell your jungler to offensively ward the enemy's jungle. The great thing about Mazlahar is that he kills minion waves very quickly and can push a lane all the way in in order to assist other lanes. He starts off slow but scales very quickly. Refer to my Malzahar guide for more details

Mid to late game:

Anivia
Karthus
Orianna

These champions usually spend the majority of early to mid game farming levels and items. Their skill set allows them to farm incredibly well while being safe from ganks. Their skills scale very well into late game which is why they can farm for the better half of the game. When I play these champs, I try not to leave my lane unless absolutely necessary and focus everything on out-farming everyone in the game.

Anivia is one of the trickiest champions to play early game. The animation and projectile speed of her autoattack is very slow which makes it difficult to last hit. There are no tricks to last hitting with her auto attacks, just practice. Anivia also depends heavily on blue buff to sustain in lane. If you miss your Q, there is roughly a 10 second window where you can be harassed for free so make sure you do not use your Q when you are in a position to be denied cs. Once you have your ult thought, you are god in lane. You can start playing aggressive, but make sure you do not run out of mana or just ask your jungler for blue buff. You can also nuke the entire wave of minions if there is a threat of being ganked to reset the creep wave into the opponents tower. A small trick I use often when trying to hit Anivia's Q is to move back slightly to make your opponent think you retreating but then quickly turn around with a Q shot. There is a chance they won't expect it and forget to sidestep.

When Orianna was first released, she was the god of all APs. Good early, mid and late game. After the gradual nerfs I believe Orianna has shifted towards the mid to late game AP role. With Orianna, I usually use her skills to kill creeps and to stop the opponent from getting in range for harass until I can buy a Catalyst. I don't aggressively harass until I have farmed enough for a blasting wand (part of Rod of Ages). After you get Rod of Ages you can easily nuke down an entire creep wave with this method:
Q -> W the ranged minions to kill them all by traveling through the melee minions
E -> Targetting yourself to do more damage to the melee minions
Q -> the melee minions once again
When harassing with Orianna make sure you first position the ball towards the side of the lane so that when the ball travels to your target, it passes through the least amount of minions to do more damage. Once mana is not an issue, position the ball between your opponent and your minions to create the "harass zone". If they get too close Q -> W them. Orianna is fairly safe from ganks. When you get ganked by melee junglers (which is most of them), don't panic. Wait until they get close to you then do E -> W to slow the jungler and speed yourself at the same time. If you still can't make it then flash. If they start backing off, remember to turn around and Q for some free damage.

Karthus has a very long range with his Q which allows him to farm safely. I have seen a lot of Karthus players rush Tear of the Goddess. It is not a bad item on Karthus but it should not be your first item. Catalyst offer more survivability and along with Karthus' passive, you rarely find yourself out of mana. You can always charge up your tear stacks later on. Karthus is the type of champion that can stay in lane to farm all day and still get fed by killing or assisting with his global ult. Remember to retreat back to your tower if you are going to ult in case the opponent harasses you for free or interrupt your channeling. If you have blue buff, you can use your E to kill minion waves quickly without mana issues.

Situational
The champions listed below are viable picks but depend heavily on your's and the opponents team composition.

Kennen
Swain

Kennen is an amazing laner because of his ability to harass for free with W passive and farming ability. He maintains great dragon control during early to mid game skirmishes and he does not use mana which allows your jungler to have it to jungle much faster. The issue with Kennen is that unlike most AP Carries, he needs to be directly in the middle of a team fight to maximize his damage potential. Without some sort of protection or AoE cc, he can be easily focused down during fights and your team will lose a major damage source.

Swain is great at sustaining in lane and surviving a teamfight. Unlike Kennen, he has the ability to stay in the middle of his damage a fight and stay alive. The problem with Swain is that his burst damage is very low. As an AP carry, if you cannot burst down a target quickly one of two things or both may happen:

Your target escapes and gets healed by support
The enemy team realizes the damage you do and focuses you down before you can kill your target

When using Swain it is a good idea to have another champion that is able to burst down targets. With the meta game moving away from super healers, Swain might become more popular. Swain's capabilities are also hampered by ignite and champions such as Miss Fortune, Tristana, Katarina etc. Champions that have skills that reduce healing. Swain should not be first picked.

Below is a relationship diagram that shows picks and counter-picks to the 9 champions within this guide. Keep in mind that this is my own opinion formed from experience laning with and against these champions (including mirror matches). League of Legends is extremely variable therefore the relationship shown below will be different. I organized this diagram for the sake of simplicity and general understanding of each champion's mechanics, pros and cons.

If they pick:
You pick: or or
Orianna - out ranges, stops approach
Zilean - out ranges, stops approach, free harass
Anivia - best counter-pick, out ranges, stops approach, free harass

You pick: or or
Malzahar - stops push
Zilean - best counter-pick, out ranges, stops approach, free harass
Swain - out sustain, stops approach

You pick: or or
Malzahar - best counter-pick, out ranges, stops push, free harass
Kennen - out ranges, stops approach
Karthus - out ranges, stops push

You pick: or or
Orianna - out ranges, stops approach
Annie - best counter-pick, out ranges, stops approach, out burst
Anivia - out ranges, out burst, stops approach

You pick: or or
Annie - out ranges, stops approach, out burst
Anivia - out ranges, stops approach, stops push
Kennen - best counter-pick, out ranges, stops approach, free harass, stuns through malz ult

You pick: or or
Karthus - best counter-pick, out ranges, stops approach, free harass
Anivia - out ranges, stops approach, free harass
Annie - out burst

You pick: or or
Karthus - out ranges, stops approach, free harass
Swain - best counter-pick, out sustain, out harass, stops approach
Kennen - stops approach, free harass

You pick: or or
Brand - best counter-pick, out ranges, out burst
Malzahar - stops approach, out pushes
Kennen - out ranges, stops approach, free harass

You pick: or or
Malzahar - out ranges, stops approach, %health damage
Orianna - out ranges, stops approach, out pushes
Brand - out ranges, out burst, %health damage, free harass

Your team composition is a very important factor when deciding the right AP champion to use. The hero you choose should not only have an edge against your opponent's pick, it must also have synergy with your team mate's champion picks. This comes from experience and knowing your ally/enemies champions' capabilities. One of the most common synergistic strategy is AoE team composition such as ( Amumu, Kennen, Renekton, Taric, Miss Fortune). All 5 champions have AoE ultimates and work most effectively when used together. Below are some examples of champions who have great synergy with each other:

with Warwick and/or Cho'Gath and/or Amumu and/or Nasus

Malzahar ult + Warwick ult
Malzahar silence + Cho'Gath silence
Amumu + Nasus + Malzahar all have %health damages and is great vs tanky teams

with Ashe and/or Alistar and/or Amumu

Ashe arrow initiate followed by Tibbers
Alistar/Amumu AoE Stun followed by Tibbers
Stun + damage composition

with Vladimir and/or Morgana and/or Alistar

Sustain composition
Great at poking and healing
Morgana black shield on Swain in teamfights

with Amumu and/or Janna and/or Jarvan IV

Orianna use E on Amumu into Amumu AoE Stun + Orianna ult
Jarvan ult inside Jarvan ult
Janna shield + Orianna shield = OP Shield
Janna speed + Orianna W
Stun and disruption composition

with Amumu and/or Miss Fortune and/or Nunu & Willump

AoE stun lock + Miss Fortune AoE Damage

with Irelia and/or Caitlyn and/or Janna

Zilean speeds Irelia into other team carries
Zilean ults Irelia to wreck more face
Caitlyn + Zilean early game domination
Janna + Zilean double global experience boost

with Fiddlesticks and/or Jarvan IV

Karthus + Fiddle sticks can kill Baron at level 12 - 14
Fiddlesticks ultimate + Karthus E
Jarvan traps people inside Karthus E

with Malphite and/or Galio and/or Amumu

Allows brand easily hit AoE combination
Brand ult bounces easily

with Trundle and/or Vayne and/or Ashe

Anivia wall + trundle wall and slow
Anivia slow + Vayne speed boost
Vayne can wall stun people with Anivia's W
Ashe + Anivia = kite forever

Once again keep an open mind when it comes to synergy and strategy. The mechanics to League of Legends is simple but the underlying strategy is deep so don't be afraid to experiment. Sometimes picking the perfect AP carry will not win you the game because you have 4 other champions on your team that may struggle vs their lane partner(s) or do not work well when team fights start to happen.

As you may have noticed, there are many other viable AP champions that I have omitted from this guide because if I am not confident enough to play competitively with them I will not write a guide on it. They all have different strengths, weaknesses and style of play but there is one strategy they all have in common: FARM FARM FARM There is no way you can compete if you do not have the items and the levels. Each champion have different tricks to farming efficiently, so who ever you play, find that sweet technique that will allow you to out cs your opponent while wrecking your lane. Anyway that's the end of the guide, Happy farming :D