Welcome to my Warwick Guide!
My Main Role is Jungle and used to play a lot old Warwick and though it's playstyle has really changed, it still remains as an offtank which can deal a lot of damage and supress an enemy during the teamfight, not forgetting that he also can Heal a lot and now is Faster than ever. Enjoy! (Sorry for any bad english mistake)

Prons:

Very Mobile.
Nice Ganks.
Good at chasing down enemies.
High Sustain at jungle and dueling against enemy champions.

Cons:

Not so tanky.
Doesn't have any wave clear.
Weak against Hard Croud Control.

We max Q at first for sustain in jungle and more damage when ganking
Secondly, our E, providing us more damage reduction.
The last is W.
We take fervor of battle as it stacks with our passive and we will be getting stacks easy while fighting enemies.

You should start with W, chase the enemy until you are close enough and then:

If you got your ultimate: E for the shield to start charging, R from the furthest distance you can reach, press Q and keep it pressed, then explode E and follow with AA.

If not: Just do the same as before but you'll have to start with your E, then Q, explode E and AA.

Passive: Damage bonus and Healing when low HP

Q: Two ways of using --> Tapping Q and releasing instantly bites the opponent, healing you and damaging them (% HP damage)
Not releasing Q button makes you attach to the enemy champion, and appearing behind him
This can be useful ganking as you can Activate your E before using it, and when you're behind the enemy, explode your E, causing him to flee to your teammate's side. Also can help you deal with, for example, a Ezreal using his E to escape, as you will follow him even if he does it.
However, if not used wisely, this can end up in getting bad positioned in a Team Fight or getting hit by turrets when ganking, so if you are sure the enemy is going to die without going backwards him, just bite him.

W: Not too much to explain if you have played the old Warwick, the passive allows you to run fast and hunt low HP champions, becoming useful even for moving around while jungling
When activating it, you can chase down all enemies in a range, even if they are Full HP.
This is the key of your gank, as you will get to your enemy before he can even know you're coming for him/her!. Anyhow, remember the Ultimate-like cooldown the ability has and don't use it if not necessary.

E: A shield which provides damage reduction and makes enemies flee after exploding

R: A skillshot which has more range according your Movement Speed, supresses the enemy for 1.5 seconds (Less than old WW's) and heals you for the 100% of the damage you deal.
With high movement speed, a hecarim's chasing power can be now evened by Warwick, as you'll have a lot of resources which will make you run a lot and cast your ultimate from an extremely far distance.

I start at Blue Buff. Depending on your playstyle, you can reach lvl 2 and instantly go for a gank, or else, you can farm the jungle and gank at lvl 3, It's not recommended to do the birds at low levels, as you don't have a good clear wave and your damage is still low, and you'll lose lots of hp for nothing.
As I would say with any champion: Do not only concentrate on farming or ganking, you should do the two of them equally. And do not ever forget dragons' timings. The most common mistake I see in low elo junglers is that they don't care at all about dragons, the thing they don't understand is not only the buff the they provide, but also the credibility and respect you gain as a jungler, not only for providing kills for your team, but also showing them you care about all of jungler's objectives. Plus, of course, having dragons gives a slight advantage at all your team's lines. Not anyone will blame the jungler for losing if they know you are trying your best to win!
And always remember to steal your enemy's amumu jungle!

In teamfights depending on your tankiness and wheter your team or the other has the advantage, there will be different timings in what you're supposed to do, but your duty will nearly always remain the same: Kill the Apc, Adc, or fed enemy jungler or top (Anyone who's dealing a lot of damage and making it difficult for your Apc o Adc to get close to the fight and win it. Rushing with W and Ulting your target till you kill them will in the better of the cases win the TF, but there are going to be special occacions where staying behind and waiting for someone to jump on your ADC and suppresing him so your teammate lives will be necessary.

You know who you're fighting, if you're against a team with a lot of CC (For example), building full damage is going to end up in you losing the game, even though you probably where fed in the first 20 minutes. Not only choosing the perfect champion for that game is enough. You should always think of hypotetical situations that may happen during the game, and build according to what you think its the best. There are a lot of games where assasins and marksman become useless against a tanky team, you should never underestimate an enemy team just because yours got the advantage at that moment. You should always think which team's got a better TF and how to nulify theirs, and in most cases your build combined with a smart playing will be enough to counter it.
If you've read till the end thanks for doing so, and go pick WARWICK and have some fun :)