Hello everyone! My name is Alex, IGN Urgodzilla. I play on EUW and I managed to get diamond with Urgot in season 10. I'm playing this champion since season 8, I had just ignored him completely about 1 year after his rework but then I accidently watched some YouTube videos with Urgot, decided to try him and I'm still playing him :)
I'm looking forward to share my experience with this champion. Also I'll probably make guides for Garen and Mordekaiser.

I find Urgot to be a pretty solid pick. He has some difficult matchups (like Fiora or Jax) but if you are good enough you can still handle it. Urgot is a pretty unique champion: he is ranged bruiser (AA range is 350) but his playstyle is actually like he is melee champion, so I'm looking forward that one day Riot Games will allow him to purchase Sterak's Gage and get full value of Conqueror's healing but Conqueror is getting changed pretty often, so I'm not sure. Also I have one suggestion: right now attack speed is pretty useless on Urgot, how about giving him extra W shot per second for every 25% bonus AS FROM ITEMS ONLY? For example, if you have Wit's end, your W will shoot 5 times per second instead of 3. Almost the same as Garen's E but Garen can get spins with AS from items and levels, while Urgot will get it with items only. Let's discuss it!
Pros&Cons

Pros

+ Good damage output, very nice DPS with maxed W Purge + Good amount of CC + Good in killing both squishy and tanky champions + E Disdain allows you to negate CC if you use it properly + Your build and skillset makes you pretty hard to kill

You are strong and fat bruiser who is good both in 1v1s and teamfights.

Cons

- You are pretty slow - Your E doesn't allow you to pass through terrain - Your range is not that great (except R Fear Beyond Death) - 3 of your abilities are skillshots which are not that easy to land - Large cooldowns and manacosts in early game

If you waste your E Disdain and miss your abilities in general, you will be kited pretty hard.

Let's have a look at Urgot's abilities:

1. Passive Very strong passive, 6 shotguns on 6 legs, each one deals additional physical damage flat+%max health to every enemy unit in a cone, scales well into late game. Shotguns are triggered by autoattacks or W Purge. Cooldown is 30 sec per leg early game but it goes down later: 20 sec lvl6, 10 sec lvl9, 5 sec lvl11 and 2.5 sec lvl13. So lvl13 is your biggest powerspike for 2 reasons: 1) Minimum passive cooldown per leg; 2) Second ability is getting maxed.

2. Q Missile. Explodes after a short delay, deals physical damage and slows everyone in a small radius for 1.25 sec. Manacost is 80 at all ranks, so use it wisely. Usually you max this ability second but it makes sense to max it first when you play against Gnar/Jayce/Kennen/Teemo

3. W Your bread and butter ability. For 4 seconds you slow yourself but get additional slow resistance and ability to pass through units (which is really important because Urgot's hitbox is huge and you are getting blocked by minions very often). You can't use AA but automatically shoot at the nearest target 3 times per second, locking onto enemy champions you've recently hit with other abilities. It works against turrets and wards. This ability helps you to instantly stack Black Cleaver and proc Press the Attack. At max rank this ability becomes infinite with 0 manacost but sometimes you should switch it off because you can accidently proc blast cones, scryer's blooms and honey fruits whrn not necessary. Each shot is a pseudo AA, which procs onhits for 50% of it's value but for some reason you get the full damage from Bramble Vest and Defensive Ball Curl. Bramble Vest is a very good counter to Urgot, I recommend to build it every time you play against him. Don't forget that W Purge range is slightly bigger than your AA range.

4. E Very powerful ability if used properly. Gives you jump (can't pass through terrain), damage, shield, flips your enemy and stuns him. If you time it perfectly, you can get rid of CC like Rocket Grab, Apprehend, Counter Strike, Facebreaker, etc. Doesn't help against suppressions.

5. R One of the strongest abilities in the game. On first cast you throw a long-range drill, which deals damage and slows the target based on its missing health. If the victim's health drops below 25% of max health you can cast it second time. Second cast throws chains on the victim (making it untargetable) and pulls it into the grinder (second cast takes 1.5 sec). Maximum time between fisrt and second casts is 4 sec (second cast will go automatically after 4 seconds if victim is below 25% max health). Upon successful execution you fear everybody around for 1.5 sec, which is devastating in teamfights. Execution is not stopped by shields but it can be cancelled or blocked by certain items and abilities: Quicksilver Sash, Zhonya's Hourglass, Ragnarok, Unbreakable Will, Remove Scurvy, Battle Roar, Riposte, Wind Wall, Alpha Strike, Hallucinate, Sanguine Pool, Playful / Trickster, Unstoppable Onslaught, Cease and Desist. I was really surprised to discover that Unstoppable Force and Bellows Breath don't save Malphite and Ornn from being executed.

Q max or E max second?
I prefer to max Q Corrosive Charge second because in my opinion rank 1 of E Disdain gives you everything you need from this ability. But E Disdain max second is also decent, because it gives you more shield and reduced cooldown.

Rune page I currently use:

Keystone. Press the attack You can stack it instantly with your W Purge.

Tier 2. Triumph Helps you to survive and gives you some extra money.

Tier 3. Legend: Tenacity The only viable option here: you don't need attack speed or lifesteal that much.

Tier 4. Coup de grace Helps you to bring people to that 25% Fear Beyond Death threshold. Last Stand is also good. It doesn't really matter which one you take, they both feel pretty equal.

Secondary 1. Demolish A lot of people would take Shield Bash here but I believe Demolish is the best choice as long as turret plates exist.

Secondary 2. Second wind Very good against poke but feel free to swap it with Bone Plating if you find it necessary.

Stats:
In stats we take double adaptive force because we don't need attack speed on Urgot + armor or magic resist depending on matchup.
Other options:
Conqueror - I don't recommend to take it anymore.
Grasp of the Undying - I've seen it several times but it's not worth.
Phase Rush - might be viable actually.
Unsealed Spellbook - only if you are 100% sure about what you are doing!
Manaflow Band plus Transcendence - good secondary options if you want to max Q Corrosive Charge first.
Taste of Blood plus Ultimate Hunter - also decent option for secondary tree.

Flash Mandatory. Disdain+ Flash combo is very powerful and will secure a lot of kills.

Teleport If you fall behind this is your only way to stay in the game. You can also use it to make some plays on the map.

Ghost It might be good to compensate self-slow on your W Purge but in this case you should replace either Flash or Teleport. I find them both mandatory but you can try it if you want.

Ignite It's overkill, you don't need it.

For starting items - see my notes at the beginning.
Here I'll try to speak about general items.
Mythic items:

Goredrinker This item is currently my favourive mythic item on Urgot. It allows you to survive in very tricky situations and you can get a lot of ability haste because of its mythic passive. Good combo with Spirit Visage.

Stridebreaker Not viable anymore since dash has been removed. Don't build this item.

Duskblade of Draktharr Very fun item because of stealth and very strong because of its stats, mythic passive allows you to get a lot of ability haste just like Goredrinker. But I don't recommend to build it unless you are super far ahead.

Eclipse Decent option, especially with armor penetration from its mythic passive. The problem is that this item is weaker on ranged champions.

Divine Sunderer Sheen items are not good on Urgot but this one is kinda decent. Mythic passive gives you armor penetration just like Eclipse.

Kraken Slayer Used to be fun and even kinda strong but now this item is no longer viable on Urgot because its onhit damage now receives 50% penalty from W Purge. Also attack speed and crit chance are useless for us.

Sunfire Aegis This item is stronger than old Sunfire Cape, ability haste from the item itself and its mythic passive feels good. Solid option if you want to be a tank. UPD 31.01.2021: Mythic passive swapped with Turbo Chemtank.

Turbo Chemtank I liked to play Urgot with old Righteous Glory, so I tried this item, feels even better because it gives you both armor and magic resist and a little bit of extra damage due to burn passive. UPD 31.01.2021: Mythic passive swapped with Sunfire Aegis.

Legendary items:

Black Cleaver Feel free to build this item first every game, it feels like mythic on Urgot. Stats and gold efficiency are amazing + you can stack armor reduction faster than anybody.

Blade of the Ruined King Each W Purge shot will do 3% current health additional physical damage onhit. A little bit weaker than it used to be but might be a good option against people who stack a lot of health ( Sion, Cho'Gath).

Wit's End Build it against multiple champions who stack armor ( Malphite, Rammus), you will be surprised how good it is. Now this item also gives attack damage but no longer gives healing.

Guinsoo's Rageblade Crit chance is useless for Urgot but this item allows you to transform crit chance into additional onhit damage. Don't forget that onhit damage is getting reduced by 50% when you activate W Purge.

Maw of Malmortius Usually I don't recommend to build this item, do it only against full AP team comps. In S12 this item seems to be stronger than it has been before.

Edge of Night Spellshield works really well against poke team comps ( Nidalee, Varus).

Youmuu's Ghostblade You can build it first item if you are stomping your lane opponent but it's tricky. Much better since Riot Games returned ability haste on this item.

The Collector Although crit chance is useless on Urgot, this is an insanely strong item. Also gives you additional gold on champion kills.

Guardian Angel Good option if you are fed and don't want to give your enemies that juicy shutdown.

Chempunk Chainsword New AD-antiheal item, you should build this now instead of Mortal Reminder because Mortal Reminder is now a crit item. Very good against Aatrox, Dr. Mundo, Maokai and Vladimir.

Serylda's Grudge Kinda replacement of old Frozen Mallet, works with passive shotguns + 30% armor penetration on top of that. Very good against tanky melees with low mobility ( Garen, Darius).

Death's Dance In S10 it used to be a core item for Urgot, now it is situational.

Seprent's Fang Completely new item, very interesting antishield option. But remember: you will not do bonus damage against enemy champions protected by Black Shield.

Silvermere Dawn New AD-anticrowd control item, you should build this now instead of Mercurial Scimitar because Mercurial Scimitar is now a crit item. Very good against Malzahar, Mordekaiser, Skarner and Warwick.

Ravenous Hydra No longer melee only, omnivamp and ability haste are really good but unfortinately its passive effect doesn't work with W Purge. It's either a bug or pressing W Purge really seems to be ONE spell cast instead of counting every shot as an AA (fixed, now works properly).

Titanic Hydra No longer melee only, heath and damage are fine and it really works on Urgot but remember that this item's effect is getting reduced TWICE. First - because Urgot is ranged, second - because of W Purge. Don't build this item first, build it second or even third.

Sterak's Gage No longer melee only. Feels like new core item, really solid.

Manamune It takes eternity to stack Tear of the Goddess on Urgot but Manamune is getting stacked really fast with your W Purge, so if you decide do build this I recommend to start with Caulfield's Warhammer. Once you get fully stacked Muramana you become insanely strong. UPD 30.04.2021: Actually in S11 you can't stack this item with AA and W Purge. It makes this item worse for our dreadnought but if you manage to stack it somehow with your abilities, it shoud be strong.

Gargoyle Stoneplate Build this if you are the only tanky champion in your team. It will help you to survive in teamfights. No longer increases your bonus health but gives you a huge shield and ability haste. Definitely feels better than in S10.

Warmog's Armor Good synergy with Spirit Visage and Gargoyle Stoneplate. Build it if you have no tanks in your team.

Frozen Heart Built it against full AD team comps and they will desperately cry trying to kill you. New passive effect shared with Randuin's Omen is nice.

Randuin's Omen Build it if you are getting smashed by crits. New passive effect shared with Frozen Heart is nice.

Thornmail Build it against Darius, Fiora, Jax, Tryndamere and anybody who builds Blade of the Ruined King.

Dead Man's Plate Supersonic speed + additional damage. Feels good.

Abyssal Mask Decent option now because if you hit your enemy with E Disdain, enemy champion will take 10% increased damage from any source for next 4 seconds.

Spirit Visage I don't really like the changes of this item (2900 cost instead of 2800? 40 magic resist instead of 55) but this is still your general magic resist item. Good synergy with any healing. Now also amplifies your shield.

Force of Nature Kinda replacement of old Adaptive Helm but works in a different way. Highest magic resist item + some supersonic speed.

Boots:

Plated Steelcaps Take it against AA-based champions.

Mercury's Treads Take it against CC.

Boots of Swiftness Might be a good option in some rare cases.

Feel free to discuss and ask any questions!
If you find anything you consider to be a mistake - tell me!

UPD 30.04.2022:
Guide has been edited a little bit for S12