...shouldn't exist. Some of them can screw up SO MUCH and still need to do SO LITTLE right to become killing machines that it is just ridiculous. Where is the balance in dominating your lane, killing your enemy 3 or 4 times and still losing completely the control of your lane and being unable to engage anymore because you were killed ONCE after all that?
Adanim2 votesDiana is probably the biggest offender. I have yet to see a Diana NOT snowball after one kill in my 3 years playing League.
LissomMars2 votesThe only ones I have seen that may carry hard is leblanc lol. And I always ban her. So Yeah lol
Not Katarina1 votesBan ALL OF THE LEBALANCED
LissomMars1 votesI would like to know what champion(s) you are talking about.
Señor Karthus1 votesThere are several Champions that snowball hard. Riven, Vayne, Katarina or Draven come to mind. Akali used to be too but right now I don't know how is the state of things with her after the nerfs.
Juggernaut6282 votesThere are several Champions that snowball hard. Riven, Vayne, Katarina or Draven come to mind. Akali used to be too but right now I don't know how is the state of things with her after the nerfs.
Honestly they didn't need to buff Akali in 5.9. Just because nobody plays her doesn't mean she isn't strong. She still 2-shots your carries leaving the tanks useless, and by the time she can 2-shot carries, the tanks will fall in just a few ults and a Q.
In my opinion though, Garen, Darius, and Volibear snowball extremely hard. As for Diana, I've been able to 1v1 her when she's fed, but all depends on your champion and how far behind you are compared to them.
Not Katarina1 votesI don't know where the Akali buffs even came into mind at Riot. She wasn't seeing as much play because she was a bit harder to use and had more counterplay, but that doesn't mean she wasn't powerful. Her mark was just harder to trigger pre-6. 
Not Katarina1 votesRiven? Yes, tons of mobility and cc in her kit as it is, so giving her a kill to increase her damage output allows for snowballing.
Vayne? Not sure about that one. Her and Katarina are both very risky. They do a lot of damage when fed, but all it takes is one person to jump on them when they waste their escape. (Vayne's Q/Flash is harder to burn since it has a lower cooldown than Katarina's E/Flash but I wouldn't consider either of them broken, since they're so squishy and it takes one bad move to end it!) Draven? Definitely a big snowball-y champion, lots of early lane harass and it's hard to not die to him, but once you get past that if he can't cash in on his stacks then he's pretty useless. Oh, and never run up to a Draven with 2 spinning axes lol. I learned that the hard way. 
Personally my biggest complaint has to be Diana. God, she's a nightmare in midlane, and one kill can turn your whole lane backwards. Cho isn't too fun to deal with either I guess, but I don't feel like he snowballs too hard compared to Diana.
Pæan1 votesBasically any champ that has either HUGE burst or has resets that lead to more kills. That's how I feel anyway.
Not Katarina1 votesBAN ALL OF THE ASSASSINS
DOWN WITH THE MIDLANE!
LissomMars1 votesWhat rank are you? lol. I haven't seen a lot. Some yes. I can agree with that. Others... Well yeah lol... You have to be Plat and higher I guess lol
KING ATTILA1 voteshahahahahahahahahahaha
Lugg1 votesTotally agree! That one kill could have come from a lag spike and had nothing to do with skill at all. Every assassin in the game and Riven, Fiora, Irilia, Cho'Gath, Rek'Sai, and Tryndamere all scale so hard when up just 1 kill.
Kwon Ri Sae1 votesAkali E buff was good. Ult change was a nerf.
theragingnerd1 votesAkali E buff was good. Ult change was a nerf.
except against vel'koz
Not Katarina1 votesI feel like every champion snowballs with kills. And it stinks, it really does. But in a sense the game wouldn't be very fair if it didn't work this way. I know it blows when like your challenjour Wukong top goes 0/4 in the first 12 minutes and you're left to deal with a 4/0 Riven who starts ganking all the lanes but if you didn't get stronger off of kills the game wouldn't progress too much. I hate it, but I find it fair in a sense. Most of them are stoppable though.
I still get your point, nonetheless. i.e The 0/4 Wukong gets one kill and is suddenly all over the map and legendary is a better example of what you're talking about. But screw that 1/4 Wukong, he fed Riven man. ;; -I'm not salty-
Champions like 
(I don't know give me snowbally champions) 
still go down very easily despite being fed, as long as you can catch them with a little cc.
But champions who get fed and then can escape/(burst you down 800 times over again because their cooldowns are -320438) if they can't run away are irritating.
(Fizz actually isn't too bad after his rework, only been against a few though) 
(NYEGH LATEGAME) 
(NYEGH LATEGAME) 
-puts Zed in here because I dislike him-
LissomMars1 votesI feel like every champion snowballs with kills. And it stinks, it really does. But in a sense the game wouldn't be very fair if it didn't work this way. I know it blows when like your challenjour Wukong top goes 0/4 in the first 12 minutes and you're left to deal with a 4/0 Riven who starts ganking all the lanes but if you didn't get stronger off of kills the game wouldn't progress too much. I hate it, but I find it fair in a sense. Most of them are stoppable though.
I still get your point, nonetheless. i.e The 0/4 Wukong gets one kill and is suddenly all over the map and legendary is a better example of what you're talking about. But screw that 1/4 Wukong, he fed Riven man. ;; -I'm not salty-
Champions like
But champions who get fed and then can escape/(burst you down 800 times over again because their cooldowns are -320438) if they can't run away are irritating.
I go Fizz mid and win 9/10 games. lol. I know he got weaker but that fucker is still broken xD
Not Katarina1 votesReally? Dear God, lol! I've only been against 2-5 of the post-buff/nerf Fizz (idk) in my 135+ Katarina games and I haven't lost lane to any of them.. I used to get my ass kicked by the old Fizz, though. I'm probably just getting pit against really bad ones. I mean they didn't hit an ult on me the whole game. LOL
LissomMars1 votesOh lol Then of course you're going to win lol. Ult does I think 20% more damage xD. Plus fizz's ult cancels kat lol. But props man. Definately. Kat is OP once you get going lol