I wouldn't spend hard money on this game especially if you're going paycheck to paycheck.
But, Karthus, kennen, kassadin, cassiopeia, in that order.
Karthus is an extremely fun champion with his ult and walls. You will wreck **** up even if they focus you, because of his passive. He lays out tons of sustained damage and can carry really hard. The only thing you have to do with karthus is not die early on (in your lane) and farm. That's literally all you have to do. The kills will come to you (with the ult) and later in team fights. If you're good with his Q you can take out some lane opponents as well.
Kennen wins lanes so hard you can go top or mid, so he's easier to place on a team. He also farms really well. He lays out tons of aoe damage and is one of the less squishy casters, or can at least be built that way without sacrificing damage. With Kennen you just win lane, aoe wipe the enemy team. You can carry really hard with Kennen.
Kassadin is a mage/assassin. His early game is slow and some lanes will be hard for you. Sometimes it is difficult to farm. However, he accelerates dangerously fast, once he gets his ult at level 6 he is extremely mobile and capable of nuking hard with Q + E, not to mention the CC they provide (massive movement speed debuff and silence.) You can carry pretty hard with Kassadin just blinking around nuking **** and picking up kills and assists. If you get fed it's pretty scary because you can basically go anywhere. Imagine laning against a kassadin who picks up a kill or two early, everytime you move into the minions he will blink to you and take out half your health. He's really scary right now in solo queue.
Cassiopeia is less seen but she's similar to Kennen in many respects. She farms well and can win her lanes if you land the skill shots. You can easily pick up kills once you get your ult, and you can force them out of lane very quick. She sustains large amount of aoe damage, and in solo queue you can carry VERY hard with her. The big problem is that she is super squishy, and if you start getting focused, or the jungler is camping her lane, it can be very hard to survive, since she has no mobility powers and her abilities are offensive in nature.
I would pass Brand and Xerath, they are great champions and can win their lanes, but they are not capable of carrying very strongly, for whatever reasons. Some people will tell you otherwise, but they don't know what they're talking about when it comes to AP. For a fifth pick I would personally take Annie or Morgana over them, Annie can win her lanes if you are aggressive and she can carry really hard. Morgana can't carry very hard but she's pretty solid in lane and easy to survive and support.
I personally run Kassadin, Cassiopeia, Gragas, Annie and LeBlanc when I go AP.