AP Singed is the very best sort of poison-caster, he poisones enemies, and make damage (a lot).
He's not a nuke-caster or better: he can't kill as fast as Cassiopeia!
He's damage-over-time what means, you need to use your poison on one enemy several times, or during a long time.
See how I play:

In earlygame, you should only lasthit minions with your basic-attack.
Your main damage source is your turret (fortify) and your lane-partner (if you're not solo).
You should stay very defensive until you have enough HP through Rod Of Ages.
It's very hard to get level at the beginning, so you MUST NOT EVER get 0/1 with AP-SINGED.
It's the very most important part of it.
All you need to do is, stay defensive, kick some enemies back, don't get ganked, just don't get killed before you have a kill!

In midgame it's time to KILL!
Play as offensive as you ever could!!!
Don't die too much!
You need to make damage, and kill much!
You have to tell your team, you're not a tank, you need freaking kills!
Just get Rylais and Rabadons as soon as it's possible, so you need money.
Thanks to your runes, you should have enough Mana-Regen, so you can start at lvl 8-9 to farm minions with your poison.
Don't waste your mana to minions.
Try to use your poison careful against minions.
You SHOULD waste your mana against champions (except thy are any tank, or maybe Soraka).
Just killsteal, you should get many kills in midgame, because after getting your Rylais you do make damage, lots of damage!

If you do everything right, game is finished after egtting your Rabadons.
If not, please don't play such offensie anymore.
In lategame, AP-SINGED gets hurt really hard, almost every class in game makes him epic damage!
And AP-SINGED doesnt make as much as in midgame.
So play careful, push a lot, and let the feed to the carrys!
If you reach lategame, game kind of sucks!
So really try to only fight 1v1, or 2v1 2 of your team against 1 of enemy team.
If you play good, you should win the game, even if you suck.
If not game is lost in lategame, so let winning to your teammates, they are the imortant center of attention!
Make sure you play with 1 or 2 carrys, or a good jungler!
Don't be too defensive, you can risk much, I just want to tell, that the enemies can give u hard damage in lategame!

If you go solo-top, it's much much easyer to play good with singed, you get high level soon, and can beat enemies, before they are too feeded.
Just kick the enemies in front of your turret, and use fortify!
Don't go to near into the enemies, then you can't do anything wrong!

If that pity case happens, you should buy something against him!
If he's AP, buy Force Of Nature and Abyssal Scepter (instead of Zhonyas and Archangel), if he's AD buy Zhonyas Hourglass and Thornmail.

I hope you have a lot of fun with Singed, and make your first Singed-Pentakill soon.
If you want you can add me in game, my Summoners Name is SherKing, but I won't answer your request, because I have 300 friends already xD.
Have fun and good luck, hope you liked my guide, please give me thumbs up!