Alright, home and got the replay pulled up. Wasnt able to get the ROFL file to run (it crashed my client and forced it to patch for a while after, but I also suck at tech sometimes, so who knows), but there was only one 9/4/11 loss in your recent history, so I reviewed that one. This is stream of thought as I watch, so sorry for how wordy it is.
Laning:
This was one of your best csing games (6 cs/min avg), but theres still room for improvement there. Your first cs misses are because you're moving out of position though, which is definitely avoidable. Point to watch for: I would guess you're tunnel visioning on the lowest health minion to make sure you land your last hit. Especially as a 1 trick where you know your last hitting damage/time well, you want to at least be aware of your next cs, so that after you hit the first you can begin the pattern for the next one.
You traded very well at level 2, got electrocute off (I think some kat's are running conq with death's dance to do ridiculous healing, would recommend checking out Kat Evolved and even Scarra (when he's not memeing) on that), and took almost no damage in return. Didnt panic into veigar wall. If every trade goes like this, easy lane win.
Mid level 2: You walk up to veigar and q him out of nowhere. This is very bad, for a few reasons. He hits a q in return, which hurts you more, and gives him free ap (20 gold effectively, i think i saw somewhere). It also draws minion aggro, since your attack is single target (like an auto) and his isn't. This means that the minions (3 casters, 1 melee, and 1 cannon) all hit you for a few seconds after the trade, while your minions hit Veigar's. Not only does this hurt you a lot, but it also makes your wave push from where you almost had a freeze, and that's not a lane you want to push (he farms decently under turret and scales, and his jg is WAY scarrier early than yours). Improvement point: Be concious of minions when trading. If you can free q for no return q when theres just a minion or 2, go for it. But take care of the wave, your q will always draw minion aggro if a champ is the first target.
Level 3: good trade again. Started q off of a minion, went in to proc damage, dodged till you can leave. Not going to point this out every time, but in a heavy critique post, I do think that positives should be pointed out to be reinforced while mistakes are fixed.
Towards 4: Several cs missed and free harass taken, I think because you were trying to manage the wave, possibly trying to setup a trade, not 100% sure. It was definitely tunnel vision that caused the mistake though, keep your head up to not only enemy minions dying, but also allies (you tanked 5 minions after getting a w to the face, and lost cs to turret as it pushed). Improvement point: When the wave is getting fucky, don't hurt yourself trying to fix it. Instead, let it shove to you, and spend your time trying to make csing under tower easier so you can get more then. Play the slightly longer game.
Same stuff to 6, fight at 6 wasnt bad, although it could have been a kill. You started the trade well, doing the exact same trade as before, but walked into the wall. He'd already missed some stuff, so no big deal, but it delayed your damage. When you went back on him, q and w had 1 sec left each, no reason not to w him and either force the flash or just sit there autoing him. You've got double long swords, he has a tear, You win that, As long as he has flash though, your ult wont kill, so you want to force his flash if possible (since you already forced the cage with a good trade). Could possibly have followed his flash with yours since you had it, but low vision of jungler coulda been sketch, no blame there. Cs after, missed like, half a wave + cannon trying to push fast. Dont starve yourself to deny an enemy if you dont have to. Improvement point: Don't rush your fights, play to your win conditions (forcing cage and flash), especially into ability reliant champs like veigar. Once you outplay them (cause you did there), make sure you dont outplay yourself too!
Throughout all of this, a few notes. 1: Your bot lane is getting trashed. Unfortunate from the start, but also makes roam opportunities. Enemy bot pushed a lot. You also had a pink in your inventory for like, 4 minutes without river warded. No problem of getting ganked since your wave was put well, but it does help your jungler gank you if thats out.
Level 8 fight, great. Positioned well before it, played to your strengths, ulted when he had no saves. Greedy for plates after tho, i think hec shoulda killed you. Take care of no jungler vision.
Annie suicides for a 1 for 1 and lets her adc die bot, nothing you can directly do, but if your team is being morons, ping them off. Lie if you have to 'He has flash, don't dive!'
Mid-Game:
Level 10, you picked up gunblade, came into lane where you had an item and he didnt, immediately killed. Good play. Another option woulda been to use that strength bottom tho, free double on luc and thresh not only woulda been more gold for you from shutdown, but also would let your trist back in the game. Especially after a turret is down, camping the hell out of bot is very viable when the enemy pushed (happens a lot in low elo) especially when your supp has cc.
As it so happens, that exact situation came mid, and you were able to pickup a double as your cds came back. Improvement point, keeping an eye on free kills around the map (another kind of tunnel vision)
Level 13: Drag coming up, team doesnt have control of it, but waves are even. You never even move towards it. No pings, nothin. You're 4/0/6, with better cs than your laner or enemy adc. You want to fight CONSTANTLY if your cds are up (everything but flash was). Ping the hell out of your team not just that its up, but that you're on your way there. Put vision out! (you have no wards, but you can escort your underfed annie so you can murder everyone who comes for her). If you want to climb, its your job to make the good call (at least 2 enemies pinged drag and forced their allies to come to it, only kayn pinged it on yours and no one responded as yall farmed mid). Improvement points: Watch for objectives becoming available, and use your weight to make the team do good things.
Pick on thresh + luc was good heads up mid-fight. Keeping eyes up turns those picks into doubles.
Be aware of lost fights. You wasted a good bit of time walking top to try and save a dead illaoi (1v4).
Trust in allies, annie pinged like hell she was going in. She's not been playing well, but also she's predator annie. Its not like they can stop her. Position to take advantage of it, if they blow everything on her like they did, nuke everyone (she even traded herself for luc there). Same thing with illaoi like, 1 min later.
Finish free kills if you're not going to chain out, keep on them while safe. Made your adc flash for nothing.
Take care of 1v4ing. You'd pinged going to drag, and illaoi came with you, but annie and kayn pinged off because of cds. Were they right? No, i dont think so. But, thats why they're silver, they don't always make the right call. You tunnel visioned on luc and got killed as drag came up, dropping a big bounty. This is the first real mid-game mistake, and is the first that leads to losing a game, rather than winning less, which is the earlier stuff. Improvement point: Watch for when your team is in the wrong spot. It sucks, but you have to play with them until you're strong enough to 1v9.
Fight at 26 min: Starts off really well on a pick, you go in and ult and... panic. Cage went down, and you didnt have a shunpo target out. So you walk into a Vei cage and get obliterated. One of the hardest things to do in league is learn to move sideways. In this situation, by sideways i mean straight down (sideways because it keeps you in the fight, same distance from veigar). If you're feeling ballsy you could e-w-q him and woulda been safe, or you coulda moved to a safe part of the cage. Its a decision you made right at least 5 times in lane, but because you'd just gotten a reset and there was a luc alive and shit does thresh have hook and everything else going on, it goes out the window. At the end of the day, you're playing kat. If you stay alive long enough, you will kill everyone (except nasus >_>) as daggers keep dropping. As soon as you hit that ult, don't look at the enemy hp, watch for your next safe spot to jump or walk. Improvement point: remember how you dealt with enemy abilities like cage before, keep a level head, and move sideways.
29: again, free drag given. Yea it would be 4v5, but also if y'all get the jump on things, it'll still be your fight. You're too strong for it not to be, kayn too. Fight at 30 was played very well be veigar, honestly shoulda just spam pinged back to stop the bleeding. Pinging out of a fight 1 person has already comitted to can get rough on tempers, but sometimes its the call to make.
Late-ish Game
30-31 min: this is bad. This is a game winner right here, and it wasnt taken. https://i.imgur.com/zFT600o.jpg Kayn gets picked hard, dies fast, you shunpo in and..... flash out. you have a dagger landing on 4 people, hook, flay, and cage are all down. Not a single person on their team has flash, vei ult is down and hec isnt there. Not a single way for them to physically stop your ult, and you're an 8/2/10 kat. Thats a 1v4 montage play right there, and instead you ran. Improvement point, have a list of 3-5 abilities in your head of what you're watching for (hook, flay, cage, hec ult would be mine this game). If they're down and you've got a go chance (especially with 2 teammates by you), light the enemy up. I don't know if this was panic from above or not realizing your opportunity, but keeping that list can help with both.
No comment on the followup. You weren't watching the cage, went right in. 5-0 ace, I would say this is where the game starts actually being lost.
As a connected note, if you miss an opportunity and are hitting yourself after, NEVER try to make up for it. That just leads to inting. Opportunities are never forced, they present themselves. And anyone 'forcing an opportunity'