This is my build for Urgot that I have been using for quite a while to stomp on solo queue ranked carries :D

There are a couple of mastery set-ups that work well for Urgot but I like 21/9/0 the most for its raw damage and lane sustainability (which is extremely important since you MUST take an early lead is Urgot in a carry lane or you WILL lose). The 21 in offense is pretty self-explanatory since the Havoc change and the lower masteries in offense can't really go anywhere else. Although most carries at least get Awareness to get as much experience as possible while laning, Urgot shouldn't need the bonus experience to out-level or keep up with an opposing carry since he has an amazing harass to push them out of the lane (so they lose experience).

0/9/21 also works very well. It lacks in the raw firepower of the offense tree but makes up for it by having Awareness and Strength of Spirit making it a very good set-up for sustained laning. It also has some CDR so you can change your Blue runes to flat MR or MR/level and still get 4 Q's in a Charge.

15/7/8 is a hybrid of the two mentioned above by getting lane sustainability, increased experience, and a little bit of damage however stopping at 15 points seems wasteful to me since Havoc is an excellent mastery now so I suggest one of the first two.

Acid Hunter is your main damage so of course you max that first. However maxing Corrosive Charge first isn't necessarily the best sequence every game. Sometimes the enemy team won't build any early armor so maxing Corrosive Charge won't be as effective as maxing Shield (and also raising your Corrosive Charge's mana cost for little gain is especially ill-advised if you want to maintain constant harass).

I usually max Corrosive Charge first because a lot of players have flat armor runes so it is very common to see high early armor values.

Flash is the best summoner in the game for carries hands-down and as such is a must for Urgot. It can be used for surprise Ults to initiate a jungle gank or finish off a low opponent (make sure the fight is in your favor though). It is also useful for escapes obviously.

The second summoner is a matter of preference. I prefer Cleanse especially when there is an Ashe, Amumu or another game-changing stun character. Cleanse is useful in most games even if there isn't heavy stuns throughout the enemy team if your lane opponent is centered around a stun i.e. Annie. Ghost is another good option for chasing and running away but its usefulness is mostly in end-game since you shouldn't have problems chasing (with Shield slow) or kiting. Ignite is my 2nd preferred summoner behind Cleanse to counter heavy healing laners like Vlad.

Exhaust can be somewhat useful if you are going against an AD carry like Vayne but generally Urgot shouldn't be brawling in other champions' ranges. Teleport is a decent option for surprise ward ganks but that is a fairly limiting case (especially in solo queue when you can rarely count on that much coordination among strangers).

Overall my Summoner rankings would be Flash>Cleanse>Ignite=Teleport>Ghost> everything else. Ignite is equal to Teleport to me because they are both great but only in specific cases.

Core:
Tear of the Goddess should be built as soon as possible to increase your mana pool obviously. However, I usually try to get both Tear and regular Boots in my first back to maximize my time in lane. Also pick up a ward or some health potions if you have spare gold (this tip pretty much always applies when you back)

Next buy should be Manamune to get a large increase in damage output to really pressure the enemy carry (and it gives you bonus mana on auto-attacks). Brutalizer should be next for even more early to mid game pressure and finally Ionian boots to reach the 4 Acid Hunter in a Corrosive Charge mark. Brutalizer will be sold later to make room for your final item (in most cases Last Whisper to maintain high ArP). If, later in the game, you decide to sell your Brutalizer the CDR bonus from Ionian is largely less important (doesn't give the 4 Acid Hunter in one Corrosive Charge anymore) so its a good idea to exchange Ionian for Merc Treads at that point.

Late game:
The last items in the item purchase order can be bought in any order really but it is very helpful to get Bloodthirster earlier if you can avoid dying while you farm it up. If you feel that you won't be able to farm it that safely postpone Bloodthirster and grab a Banshees or Guardian. Always build according to the momentum in the game!

Although I suggest Banshees and GA as buys there are other good options depending on the enemy team:
Malzahar/Warwick targeting you with their ults --->Quicksilver (Cleanse covers regular stuns)
Heavy AS-type champs or generally heavy AD ---> Frozen Heart (extra CDR too!)

Most other defensive items are sub-optimal though they could work in extreme cases (i.e. Thornnmail).

However, for the items BT and Last Whisper should be your main late game buys nearly every game. BT will not work if you are absolutely unable to farm up the AD you can build IE (high AD but that is the only thing Urgot benefits from) or Hextech (even lower AD though has some spell vamp which is ok I guess) although if you are unable to farm because of enemy map control and ganks you probably already lost :P. Last Whisper can be replaced with another BT or another defensive item if the enemy team has extremely low armor. Youmuu's is a decent option if the enemy team isn't stacking armor with a good active and to maintain high CDR (If you get Youmuu's do not buy Last Whisper).

The best way to figure out how to play Urgot is to actually play him. Get used to his range and its manipulation especially.

Landing your Corrosive Charge is essential to being a successful Urgot so try to:
Pick up on how your lane opponent normally moves around and use that to your advantage
Watch your own creeps' health to see which he will try to kill next and punish him for it
Bluff your Corrosive Charge shots at first and see which way he moves (if at all)

Corrosive Charge can also be used to check bushes (or an Acid Hunter shot).
Watching your opponent can also tip off jungle ganks so be wary of irregular movements.
Harassing before level 3 should be done conservatively to reserve mana for your much stronger level 3 Acid Hunter harass.
Learn to skill shot your Acid Hunter well especially for kiting purposes.
If you are having trouble landing Corrosive Charge you can try to activate Shield and hit with an auto-attack or skill shot to have an easier target.

Jungle:
For Jungle buffs I generally decline both Red and Blue because Red only benefits auto-attacks which Urgot does not rely on and Manamune should cover mana problems.
Stick to killing small camps especially when you need to farm your BT

Laning:
Aim to lane against champions like Annie or Malzahar because you can easily harass them early since they are heavily dependent on their ults.
Avoid laning against champions like Rumble or Maokai because you're early damage is going to hurt a lot less on tanky targets.
Urgot isn't too great of a baby-sat AD carry (Ranged AD Carry + Support in bottom lane) because it's a lot harder to harass a champion that is being supported.
Focus heavily on last hits! Farming isn't just getting kills!
Urgot is a very effective zoner since he has early dominance and great range so do it when it is safe (meaning you have strong ward placement to prevent a jungle gank).
If you are losing your lane build an early defense item prereq (Negatron Cloak or Chain Vest)

Ultimate:
Do not initiate a fight with it unless you can flash out after the switch to avoid death
Very useful for coordinated jungle ganks
Also extremely useful for getting away from enemy jungle ganks (ult the jungler and run away or flash if needed) because it also slows the target afterwards.
Can be used to cancel channels through its suppression (Malz, WW, Fid, etc.)
The armor/MR bonus can help in 1v1 situations especially when you are about to take a large hit (i.e. a Vlad ult)