Reasons why these runs are great:
-Makes you faster
-Makes you hit harder
-Makes you hit faster
-Makes you tougher
Plain and simple, got it? Good! :D
Twenty-one points in offense will give you the damage output you need to assassinate your enemies. Offense is pretty simple: do more damage. Got it? Good :D
Nine points in Utility buys us some additional move speed, healthy mana regen and cooldown reduction to spam your abilites, and lastly a bit of an extension on all your neutral monster buffs goes a long way when ganking. We like ganking. Enemies are delicious. Warwick hungers.
Alright, so you're still reading. Good, because this is where you learn how to win. Winning is delicious.
Always start with the basic jungle package. It's the bestest.
Start at good 'ole gromp (Gromp is the delicious frog that spawns near blue buff). Gromp is kind of like the appetizer before the main course; the main course is the blue buff monster.
If your mid/bot/top lane (depending on positioning) is feeling frisky, go in for a gank. Remember your priority is to feed the lanes, but that doesn't mean you can't finish enemy champions off. Use your best judgement to decide whether to take the kill or not.
Barring ganks, continue jungling. Proceed to wolves, and from there, wraiths. On your way to wraiths, harass the mid lane a bit. Pushing the enemy mid laner away from the minion wave will give your team's mid laner a chance to last hit enemy minions without too much pressure from the opposing mid laner. At the same time, pressuring the opposing champion off their farm will deny them some gold giving your teammate an advantage in lane. This same concept can and should be applied to all other lanes while you are moving through the jungle.
After wraiths, head to the Krugs (the two rock monsters near red buff). Krugs and red buff share the same basic concept as Gromp and blue buff. Krugs is the appetizer; red buff is the main course. Smite the large Krug as soon as possible, and once the Krugs are clear, proceed to red buff.
After getting red buff, check and see if you can get Dragon. First, try to remain hidden from enemy sight on your way to Dragon. You want to keep this as secret as possible. Head to Dragon and take out the Scuttle (monster in the river) with the help of the Krug bonus. If you see no sign of enemy champions during Scuttle, it is probably safe to take Dragon. Try to position a ward in the enemy jungle to try and anticipate opposition. Try to take Dragon as quickly as possible. Remember, your masteries and runes make you faster than most champions at this point in the game, so don't be afraid to bail if someone comes to stop you.
If Dragon seems impossible after red buff, it may be best to just head back to nexus. Pick up some boots and the Skirmisher's Sabre. Head back to the jungle and start over at Gromp, sweeping across the jungle just as before getting blue and red buff as they pop back up.
Remember to constantly keep an eye on Dragon throughout the game. One of your goals as a jungler is to prevent the enemy team from getting Dragon while getting as many Dragons as possible for your own team. Usually the best time to take a Dragon out from under the enemy is to take out the enemy jungler. On top of this, smite the large wraith to get vision of wards and check Dragon for enemy wards. Doing this periodically will ensure that the enemy will not only be blind to you taking Dragon, but also keep the enemy team from having a timer on the next Dragon.
Remember to systematically clear the jungle during this period. This means either sweep from top to bottom or vice versa. Keep an eye on your teammate's lanes and gank whenever your ult is off cooldown.
If the enemy jungler seems fairly weak and doesn't have a lot of crowd control on them, don't be afraid to invade their jungle. Since you are using Devourer, you want to clear as many jungle camps as possible. Use wards to anticipate the enemy jungler and you will be just fine!
As a side note, warding the enemy's red and blue buffs can get you kills on your own. If you see the enemy begin one of the camps, try to squeeze in your ult before they have finished off the monsters. This will help you deal additional damage while they are locked down from your ult. If you don't think you can kill the enemy, smiting the large monster and running may not seem very Warwick-like, but being dead is even less Warwick-like ;)
During this time in the game, you need to build Wit's End and Blade of the Ruined King. These items together make your ult devastating and give you max attack speed (2.5 hits/sec) at level 18 when you use your Hunter's Call. If you find you need just a bit of armor (since wit's end will only provide magic resist), purchase either Chain Vest or Warden's Mail depending on what you need for your build. If you really need more magic resist, grab a Null-Magic Mantle or Negatron Cloak (if you are building an Abyssal Scepter).
At this point, you should still continue to clear the jungle to get as many stacks on your Devourer as possible while staying near your teammates for team fights. In team fights, your goal is to lock down whichever enemy champ deals the most damage so your team can collectively stomp their face in. Be wary of AP champions with Zonya's Hourglass. This is one of the few abilities (along with Gangplank's Remove Scurvy and Bard's Tempered Fate) that can cancel the lockdown affect of Warwick's ult.