Warwick is by far the best jungler in the game and jungling (aside from not feeding) is one of the best ways to help out your team. Utilizing his strengths, WW can be one of the nastiest champions out there. This is meant to be a guide for those already familiar with WW.
Summoner Spells - Teleport and Smite. Always.
Teleport is your friend, after you've won a team fight, and you are pushing a lane, teleport away and single-handedly take another turret. Game getting rough at the end? Drop a ward for a double backdoor... oh yeah, you can teleport to wards. Allies in trouble, and being chased... save the day. Team fight gone horribly wrong, well you cant defend turrets by yourself vs 3 or 4 champs, but you can teleport up to their turret and take one yourself or at least force them to stop pushing and chase you while you B to base.
I rune ArmPen, dodge, CD/R and A/S quints, although the only 2 I think are non-negotiable are Pen and CD/R - the more you can spam (W) for your team mates, the better they will do, and if you dont run CD/R, ask yourself how many times you've died because you were 1 second away from an ability to live...
Always start at the Wraiths - I have never been successfully ganked when starting there, but I have almost everywhere else. If the other team obviously doesn't have a jungler, then I move to the creeps next to the lizards, then steal the other teams Golem, and head back to the wraiths. If the other team does have a jungler go for your golem.
-Your first ability at lvl1 should be (W), the A/S bonus; this is a GLOBAL BUFF, and if at any time you see your team mates engaged in battle, from lvl 1 to lvl 18,, hit (W)
-Watch the mini map constantly, you shouldn't need to watch how you're doing vs the creeps... thats right, WW can jungle blindfolded, so appoint yourself in charge of map awareness for the team. Ping if you see something suspicious.
-At your earliest possible convenience, drop a ward in the bushes between your 2 solo lanes; if your enemies are capable, they will often try to gank mid.
-Do not return to base until you have 1050 gold or you are about to die.
After you have the Golem Buff
-If at any time your team gets killed in the lane, teleport to that lane and hold it, but make sure you last hit every creep. (you will get a lot of gold and exp, plus save a tower)
-If at any time your team gets a double kill in the lane, teleport and push the tower, you will get it 2/3rds dead if not take it.
If you have a Golem buff, you ult has a reasonably low CD, I will often run into a lane, ult an enemy and run back to jungling, it scares the enemy and can often lead to a kill for your allies later on.
In team fights, use it only to initiate if an enemy gets trapped inside your turret range or a squishy gets out of range of his buddies and your team can pounce on him quick. The 2 second stun, and ult is usually enough for your allies to unload on him.Otherwise, its best to save it until someone gets down to half health, or they have scattered and you can stop someone from running away.
Although the opportunity is uncommon, using it save an ally or even sacrificing yourself so your carry lives and keeps his killing spree/stacks is the absolute best use of your ult.
-Wards: I can not stress this enough. Before any turrets go down, there are three main places you want to put the wards, between the lanes and at their golem. As soon as towers go down, put wards by the lizards and golems of the towers that are down and always keep wards up between the lanes. There are excellent guides on ward placement, I suggest you read them. Twice.
Start with Cloth Armor, 2 Health Pots and a Ward
First trip back: Madred's, Boots, as many extra wards as possible.
2nd Trip: Boots of Swiftness (but Aixtek, Serkers is awesome! and Blood Scent...; Not as awesome as running faster then everyone ALL the time)
After that, everything is situational
If things aren't going great, and you are being pushed, I always rush Tiamat (but Aix, Tiamat are garbage; Warwick becomes the most efficient pusher in the game with a Tiamat... whats that, 25-30 creeps baring down on a turret? plow through them in 3 seconds with (W)... just killed the dragon and you notice they have a player down, and everyone else is commited? jumped into a free lane, and push to a turret, and take it in 10-15seconds. It also gives a nice bit of HP/5 and enough Mana/5 to keep your (W) spammed non-stop. BTW, your ult now does an extra 200dmg. But the greatest thing about Tiamat is you now farm like crazy, even when you lane against the other team, which helps in buying all those wards.
If things are going good, you can take your time getting Bloodrazor first, while picking up wards.
Other good items for WW
-Malady (Stark's if your team is Att Dmg heavy)/(Executioner's if they have healers)
-Guardian Angel, (Banshee's if they are caster heavy)/(Thornmail if they are Att Dmg heavy)
-If you need more speed get Zeal (both upgrades are useful after that),
Warwick is all about supporting the team, but sometimes that means triage, knowing when to run away (make sure you tell people to go back) and pushing another lane. Build yourself and play to specifically counter the other teams carries, or support your carries (if you can feed your own carry, that should be a win). Be fluid on the battlefield, that means leave behind any lanning mentalities you have, be in as many places as you can, but never for too long. Always get the dragon, tell your team to go back and defend if you have it, the gold build up for your entire team is worth it. Put wards everywhere, I spend on average more then 1200 gold on wards each game - it helps with the fluidity.
And finally, don't curse you're team mates out for making a noob mistake, instead, take a deep breath, buy 2 more wards, and diplomatically offer some SOUND advice about what to do next time.