I don't really have the time for a replay because I'm busy but based on what I see in the post game data I can point out a few flaws and some some good things.
First off the good.
You noticed the other teams heavy AD dependence and took advantage of it by building armor. You built Rylai's, I'm assuming for the slow which is a good choice and you had good farm.
Then we have the not so good.
You were involved in less than half of your team's kills with 2 players being involved in more kills than you. This is a problem because it means you were not present when these fights or skirmishes took place. And being the second most fed on your team your team needs your presence to help them win fights. I also would not have picked katarina if this was a ranked game because it has left your team lacking any reliable stuns aside from zyra.
Personal dislikes with your build.
You can Listen to what I have to say or not this is my recommendation and is up for debate on whether it will make you better or not. I don't like sunfire cape in general, low armor and mediocre Health boost that is unneeded on an AP carry, I would have built a guardian angel to counteract the heavy AD. If you were building bruiserish I could understand the sunfire cape but would recommend a randuins omen instead as it gives better everything. The boots of swiftness I also felt we're unnecessary because of your Rylai's and your team's strong chase potential as well as karthus ultimate to secure anyone who flashes away or tries to run. I think ninja tabi for bruiser build or sorcerer shoes would have been a much wiser purchase.
I hope these comments have helped you in any way and I'm sorry if I sound like I'm talking down to you or using an unpleasant tone I'm just tired and probably shouldn't be staying up on the forums using my brain.