Well, one major factor I'm already seeing is that you're playing too scared and not warding; what you want to do is ward one side of the lane so you can see the jungler coming from that side and hover around that side of the lane. You can easily back out before they can do anything. Try to ward the river bush (not the lane bush or ones leading to blue)
You also ate Karthus Qs that weren't even aimed at you. Try staying a little away from minions until you have to move forward to last hit it; try not to walk directly up to them, keep even spacing with the lane opponent (just outside of their CC/poke range) and rotate around them so you can take it safely.
You can punish the enemy laner when they go to last hit as well, since you know where they're going to be standing when they start their AA animation. Use skillshots (Tal WEQ combo for example) to make them afraid to walk up to CS. This is known as "trading stance".
Also, you got
early vs a Karthus with Teemo Jungle. You should have just rushed Morello's or even Abyssal. Going immediately defensive with itemization that won't really help you just sets you back. If Karthus would have killed you with his ult, you made a mistake and you shouldn't rely on Zhonya's to dig you out. Doing what you did is like Irelia rushing Randuin's Omen because they know they're going to make mistakes.
Finally, when USE YOUR AUTO ATTACKS MORE. There were a few times where you could have killed people if you had been auto attacking while you went in on them, but you just hovered around waiting for their potions to heal them, essentially.
I'd suggest playing champs that can more easily hard-carry through upfront damage or CC until you get more reliable teams as well. A large part of that game was Amumu feeding, so you need to be able to carry past that.

Last bit of advice for that game: Farm. Ward so you can't be stopped from farming. Farm until you hit the point where you can hard carry and then do it. Low MMR players hardly farm. Doing so gives you a HUGE leg up on them, even if they're several kills ahead.
If you want a reference, you hit 2 items at around 30 minutes. I'm often full build near that time going full AP as Vlad, a horrible early game champion.
If you get set too far behind, afk push a side lane.