One of the most entertaining games that I can recall ever playing in League of Legends came just a month or so ago. It was just after the buffs to lethality, and as such I was playing one of my favorite mid laners of all time, Varus, and I was up against a Corki mid. As this was just at the beginning of Corki's seemingly random rise to the dominant powerhouse he is now, I was somewhat on the back foot during the laning phase, but the game eventually evolved to the point where both Corki and I were stupendously, double-digit kill fed, while the rest of our respective teams were quite far behind.
The reason this game always sticks with me, though, is because it highlight an important principle: even if you're the most fed member of your team, you don't need to be a hero. Our overall team composition was Maokai, Ivern, Varus, Kog, and Malzahar. Now, come the thirty-five minute mark the enemy Corki was obviously frustrated, as he was starting to take more risks and was complaining about the state of the game in all-chat. While I can't blame him for being annoyed at getting poked out from endless arrow spam -- I admit that it's pretty annoying now that I'm often on the other end of it -- his frustration led him to make the play that made his team eventually lose. See, my team lost nearly every teamfight during the game, and our good overall KDA was almost entirely due to snipes that I got either before after the fights started, and we were thus quite far behind on the map.
Then the Corki took the Package and dove headlong into our team, looking to assassinate me and allow his team to finally crack into our base. He then promptly got hit by Twisted Advance, Exhaust, Nether Grasp, and Chains of Corruption all one after another. Needless to say, he promptly imploded and we easily mopped up his team with the 13 kill Corki out of the way, and the victory allowed us to take the freshly-spawned Elder Dragon and eventually Baron. From there, it was easy for us to win the game, all because Corki decided to throw the whole game away.
Post-game stats were clear, the only metric our team was winning in at the point he decided to charge headlong into us was in damage dealt, almost all of which was thanks to my Piercing Arrows. At the time, we were down five turret and a number of Dragons. The game was very much in their favor, but Corki decided that he was going to make a risky (read: suicidal) play to try to win the game on the spot, instead of letting the game run its natural course. For the majority of champions, such an exchange isn't worthwhile, especially if you're the sole force keeping your team in the game. Even if you're an assassin such as Talon, splitting off into a side lane is a far better idea that trying to assassinate a sole enemy in the midst of their entire team.
Remember: you're on a team for a reason. There's time for individually amazing plays, but learn to recognize when it really is time to step up to the plate and when it's just your ego leading you into a trap. Often, you won't get a second chance when you throw the game like this -- it only takes a single Baron to completely swing the game one way or another, after all. Keep your head cool, your trigger finger restrained, and, even more importantly, your team in the loop. If you see a game-winning play, make sure your team pursues it together. Lone wolfing will just get you killed, no matter how fed you are. Remember that, and you'll find LP that much easier to acquire.