Using your lead as assassin to close out a game in league of legends

While the other answers give some useful hints on how to snowball the game early to midgame with ganks and pressuring objectives after those ganks, this doesn't always result in victory, especially in low-mid tier LoL, where team cohesion is usually lacking.

A thing you always must consider in the mid- and lategame is splitpushing. Assassins usually are good duelists, they can kill quickly and they are highly mobile allowing them to escape sticky situations. A champion like Kha'zix is very weak in big teamfights even with a substantial lead, he excels at eliminating isolated targets.

Later in the game, your focus should be splitpushing. As long, as you are able to duel most members of the enemy team, you force them to respond asymmetrically. Either they send one person, who you kill, turning it into a 5v4 or allowing you to continue pushing, forcing a further response or they send more than one person, which tilts the game in your teams favor on the other side of the map. Between Kha'zix ultimate and jump, you can escape many such gank attempts.

One more thing to remember: Keep aware of the map, as the jungler, splitpushing is more risky, you are a key member at taking baron and elder and you don't have teleport. You have to make sure, your splitpushing doesn't gift the enemy team either of them, meaning you need to smarter about when and where to splitpush than you would as a laner with teleport.

You also need to think more about your back timings. At ~4:30 you were killed, while being 2/0/0. This should not happen, you had a massive advantage at that point. But the buy order reveals the problem, you bought your first items at 4:50, after you died and buy items worth 1650 gold, that entire gold advantage was pretty much worthless till then. You should have backed at 4 minutes to spend your gold and make Kindreds life hell, instead you died with your starting items and donated 450 gold to Kindred in a situation where you had a big gold advantage but an item disadvantage. So instead of being able to bully her with your two longswords, you allowed her to almost equalize your build.

How do I transition an early lead into finishing a game?

Pace/Tempo is key here. What most people do wrong with their early lead, is to feel secure and slow down, which is the worst thing you can do. Your game has to end before min 30, if you have a lead. Follow up with this list and decide for yourself, what is best at the given moment. Put your focus on executing as much as possible, as fast as possible, since you become the agressor here:

• Force dragon/herald

  Gank top or bot, trying to force the enemy lane to either die, recall, or end up too low on HP in order to contest the objective you will take straight afterwards.

• Force towers

  If possible, take the tower. Optimally, you can get the tower and follow up with dragon/herald afterwards, however mostly times short for both of it. You decide what you go for- infernal or tower, its mostly up to what you think is best. Both can give you more control over the map, thus helping you control more objectives, which then again give you more control. Snowballing itemwise is helpful, but this is snowballing mapcontrolwise, which has a much bigger impact on the outcome of the match. Your teammates can also start roaming around afterwards.

• Gank Lanes and shove the wave into the tower, while trying to leave the cs to the laner

  Obviously, but mostly when there is no objective you can take, when you cannot get a tower or any other objective, getting a gold advantage for your laners is still better than nothing else. Focus on getting your laner ahead, not entirely yourself. On many ganks, laners lose more gold than they gain, since they often have to sacrifice some CS in order to follow up. Try to leave the kill if possible and when shoving in the wave, try to leave the lasthits.

• Counter jungle/gank

  if you are wide ahead of your enemy jungler as kha and you know kindred should be a piece of cake, make her life harder. Efficient counterjungling requires loads of variables, but keeping the red and blue timer in mind and warding important spots, will make up most of the information you need to understand the jungling pattern of your enemy.

All in all, your biggest enemy when being ahead is time. Time will take your advantage away, being ahead is like a temporary buff, much like baronbuff. If you don't play it efficient, it will cease to exist. Efficiency is Work divided by Time, so make sure you don't wander around putting no pressure onto the enemies. I can assure you there is almost no limit to this. The visibly noticable difference between people, who have 90%+ winrate in gold and actual gold players is almost entirely this efficiency. Once a master player gets fed, he takes complete control over the game and will shut it down faster than you can blink, that should be your goal aswell.

Thats for answering your direct question, however considering you're searching for advice to improve, looking at you overall data, I suggest you focus on the reasons you die. Most of the time it will be situations, that could have been avoided in first place, keep those in mind and make sure you don repeat the same mistakes over and over again. Dying when fed is the best way to make sure your lead is nullified, especially since the bounty rewards make league ridiculously equal-of-outcome economywise, where you can suck however hard you wanna, 1 kill on a 3/0 kha can get you back into the game. Don't forget: 3 solokills on the enemy jungler consecutively get you ~800 gold, if you die afterwards and the enemy team even gets an assist, they get 750 gold. Despite being 3/1 and having completely outplayed your enemy, who only got lucky once, the outcome is not only the same direcly, but indirecly you lose even more, since mostly you're the one sacrificing jungling time, where you could've otherwise cleared camps. Nowadays, playing assassin is sadly all about not dying, rather than "assasinating" others. Killing has become too simple and not dying too important. A good way to start, is to look at how regularly you are aware of the position of everyone else in the game, especially the enemy jungler. Focus on increasing your minimap awareness, aswell as doing research on what information it can give you, but that's another topic, which has millions of good tutorials.

Assassins are great in soloqueue for snowballing but as you have noticed their power drops off quite a bit the longer the game goes on.

It's very easy for an assassin to get shut down by hard CC and properly placed wards. Also in the later stages of the game people tend to stay closer together which makes picking out a target harder since assassins aren't great in 1v2 situations.

If you are playing one you should always keep the following things in mind:

• The objective of the game is killing the enemy nexus. This means kills are merely a tool to make you stronger and not the main goal of the game.
• Pushing a turret after a successful gank will give your team more of an advantage than simply getting kills. It'll reward global gold and the laner can roam around to further snowball the game.
• Know which lane to gank and/or camp and don't try to be everywhere. Some champions can easily pressure all lanes at once but often it's easier to simply camp a single lane and shut them down completely. If their squishy midlaner with no natural escapes has no flash, don't be afraid to gank him 3-4 times in a row. He'll lose farm and XP.
• Tilt the enemy. I know this sounds like a dick move but it's super effective in ranked. As soon as you realize that an opposing player is insulting you, camp him even more. He'll play worse as the game goes on. You can also put in a smiley face after a kill to annoy them more but keep it at that (Please don't actually flame/insult anyone). Also don't do this in normal games. Those are for having fun and nobody likes people who provoke in normals.
• If the enemy jungler is behind and/or weaker than you, go counterjungle. Effectively shutting down a jungler will turn the game into a 4v5 which is a huge advantage. As you do this you can also use your wards more offensively which means everyone in your team has the exact position of their jungler.
• The most important thing: Play safe. In soloqueue not dying is more important than getting kills. Only ever commit if you're 100% sure you will get something out of it without dying or if you die you should at least get an objective.