IF you'd like to get to playing Warwick as soon as possible, here's the quick summary.

Wait for you laner to get into position, Hook them, burn them.

Start your jungle at blue, if you don't have a partner to help damage it, pop a potion and Q whenever it's up. Take down the wolves in similar fashion, pop W when it's up. Your Smite will be up by the time you finish off wraiths, power down the Lizard Elder and his goons and gank top/mid if you're purple, bot/mid if you're blue. Whichever one is pushed the most by the enemy team and you will give a distinct advantage to.

Drop a ward in the river along your way to either, in the bushes of your team's lane if they haven't warded or in at dragon/mid bush if they have.

Clean up the enemy, wash rinse repeat.

Tactics: If your team has engaged the enemy, don't waste your ult to engage them, run up behind and take them on, if they run, that's a good time to ult on them.

Ganking Bot: Go for the Attack Damage Carry first, not the Support. If the support is all that's in range, however. Take it. ADC vs. ADC/Support, the team will win.

Ganking Mid: Going up against an AP mid is easy, wait for them to get comfortable, push their lane out a bit and jump up on them from behind. You'll be no match for them alone until you get some Magic Resist(your masteries and runes give you armor, remember). So only gank mid if the mid laner has enough HP and Mana to take them out in the window you give them with your Ultimate.

Ganking Top: These are the tanky sorts, generally. Depending on the opposing top you'll want to initiate a fight in several different ways, for singed wait until your teammate has them down a good bit before even thinking about a gank. If your lane is being pushed hard, however. Gank top and often, even if you know you can't get a kill. The pressure will definitely make them think twice. Ask your top to ward their brush if they aren't already.

That's the basics, I'll go more into detail about the abilities next.

The standard Warwick rotation has always been to open with the ultimate followed by W, Q, Auto-attack. This build allows you to gank as soon as you get your red buff with the extra move speed and flash combo. Flash from the bushes onto a pushed lane and open with your Q. If you're lucky they'll focus you but if you're not-so-lucky they'll make a run for it which is where your E comes in. Just remember, don't chase the Singed.

Q - Your bread and butter, use this ability during every gank. Either to initiate after a flash or to clean up a good chunk of health after your ultimate. If your ultimate doesn't take them within range of your E's activation, this should. This spell only works as Magic damage, it is increased by your Ability Power(something we don't really build, the % damage does the trick against any character)- Take this First.

W - This will increase not only YOUR attack speed, but the Attack speed of YOUR TEAM. Use this as a starter in a teamfight to give your team a great advantage.

E - If a nearby enemy is low on health, you get a HUGE speed boost. The new Furor Boot Enchantment REALLY helps here too, it gives a % bonus, that adds on TOP of this bonus. Making you nearly impossible to outrun(singed or rammus can go just as fast)

R - Your Ultimate, This can be used as an opener, to hook an enemy that came too close, to stop a fleeing enemy, to steal a kill(jerk!), or pretty much for anything. It's a stun with a HUGE damage bonus. On top of that it hits 5 times and applies on-hit effects. Your Black Cleaver and Blade of the Ruined King were made for this ability. They go perfectly together, one complimenting the other.

On to Items

Start with a Hunter's Machete to speed up your clear time(Warwick's Jungle Clear is painfully slow) Grab two health pots, use one at Blue golem(leash has been taken out of the game now) and one when you are going to your first gank(if required, if not save it for dragon).

On your first return get a couple of sight wards, Madred's Razors and Boots. Upgrade to Ninja Tabi or Mercury's Treads according to team needs. If you are doing a lot of ganking early on, upgrade your boots with furor to get a boost to your already high move speed on low health targets, allowing you to take them out even under the turret.

Build a "Bilgewater Cutlass" to "Blade of the Ruined King" next and use it with your ultimate to sap their move speed, giving your team a guaranteed kill every time you gank a lane. Build a Frozen Heart for the CDR and extra survivability in team fights. Finally build a Spirit Visage and/or The Black Cleaver depending on your team needs.

Warwick's abilities still deal with Magic Damage, doing a percentage of an enemies health in magic damage. Even if Madred's Bloodrazors were taken out, magic pen is our best bet.

Offensive Tree:

Fury: Attack speed allows Warwick to build up his passive very quickly, three strikes on the same opponent gives increased health regeneration per hit.

Blast: Just to get into 8% magic Pen, really.

Arcane Knowledge: 8% Magic Pen is a ton in the early game, powering through mid-lane ganks in record time.

Defensive Tree:

Summoner's Resolve: Smite Boost for Gold.

Durability: Extra Health is always a plus, especially for a tanky health thief.

Hardiness: Extra Armor will help with early game jungling.

Utility Tree:

Summoner's Insight: Since so much of your ganking relies on your Flash, the 15 second cooldown reduction is a godsend.

Wanderer: Move speed out of combat will help you move more aptly through the jungle, lowers clear time as well as helps get to your target for a gank.

Meditation: 1 Point to get to the third tier. Enhanced Recall can be taken instead but Mana regen, however little, is more important in my opinion.

Mastermind: Reducing the cooldown of your Flash even further, this is awesome.

Runic Affinity: Increases your longevity in the jungle, allows for more ganks more often.

Vampirism: 2% Lifesteal will help you with your auto attacks, 2% Spell Vamp will help you with your magic damage, it's a win/win.

On to Runes

Magic Pen will help you get those kills, Armor will help with viability in the later team fights, CD Reduction will help with your ownly real ganking ability, your Ultimate's cooldown. Taking that below a minute is a real life-saver.