The following guide is being written for semi experienced players of whom already have a pre existing knowledge of how jungling works. It will outline your role as a jungler Warwick in three parts. Early, Mid and Late Game. I've successfully used this exact build + some slight variations (For counterbuilding purposes) in quite a few ranked and unranked matches.
For those that are not familiar with Warwick, his role actively changes throughout the game from a jungler, to a ganker, to a DPS/Assassin + Semi Support. At least this is how it would optimally work.

I'm going to save the redundant explanations of what each attack does and jump straight into how to actually use these abilities correctly in conjunction with your role.

The mistake many players make is misunderstanding a Warwick's role in a 5v5. Just because AP and Tank builds are viable, it does not necessarily mean they are optimal, the reason being, there are far better tanks and AP warwick is easily countered making it unpractical. So without further ado, here is how Warwick is correctly played....

*Yes, this build lacks lifesteal. For those that have played high DPS builds with high levels of lifesteal, you'll know a DPS Warwick dies very rapidly upon being focused rendering life steal useless. Any negligible damage taken can be healed via Eternal Thirst + Hungering Strike.

Pros:
One of the strongest junglers in the game
Great survival
Strong Damage output both sustained and in bursts
Great at ganking

Cons:
Ult can be interrupted
Dependent on Blue Buff early game

Your first purchases will be Cloth Armor and 2 Health Potion and you will select Hunters Call first.
It's possible to successfully jungle without any health potions at all, however, this will leave you constantly at a lower than optimal percentage of health. If you absolutely need to gank or need to hold a lane, you will be greatly disadvantaged.

Jungle Route:
1.To begin with, have an ally champion leash Ancient Golem. Kill the two smaller lizards first, immediately Smite the Ancient Golem and after 3 melee hits use Hunters Call. This will maximize health returns. Use your first Health Potion now.
Keep an eye on the time. The Ancient Golem respawns in exactly 5 minutes. Take note of this exact time.

2.Providing your ally left the vicinity before you slew the first two lizards, you SHOULD be level 2. Select Hungering Strike now and head straight for the nearby wolves, kill the smaller wolves first, leaving the larger one last. Use Hungering Strike on the first wolf to kill it quickly and reduce incoming damage, use Hunters Call and Hungering Strike virtually on cooldown to maximize health return and clear the jungle faster.

3.Head for the Wraiths now. Similar strategy, use your abilities on cooldown killing the smaller Wraiths first and the large one last.

4.Head for the Twin Golems now. Open with Hungering Strike, Hunters Call and a Smite on the same Golem. After both are killed. You should be level 3 and can probably afford your Madred's Razors. Recall now, heal up, restore mana and get your razor.

5.Once ready, head straight for the Lizard Elder. From here on out, the jungle should begin respawning.

6.Work through the jungle as it respawns, keeping an eye on the remaining duration for your Blue Buff, upon wearing off, simply play conservatively, using Hunters Call on cooldown and Hungering Strike if health starts to drop or you need a kill.

Tips/Extra Info:
Your experience as a jungler will vary greatly from game to game. Sometimes an enemy jungler will try to counter jungle, other times you will receive bad RNG with low proc rates for your Madred's Razors, and other times, you'll need to hold lanes for allys or gank before level 6. What seperates the good Warwicks from the bad are those that can plan ahead and work around this.

If your jungle is crippled via an enemy jungler, use wards to try and gank/kill him or her. Sit in the top lane until it begins respawning. If you need to back for health, don't be afraid to do so, just try and do your best not to have this happen too much.

As a general rule, as soon as Madred's Razors is done, buy Boots and later Sorcerer's Shoes as soon as you can afford them. They make jungling alot faster.

Upon gaining Infinite Duress you can now gank lanes, and with Warwick this can almost always lead to a kill or at least an assist.
In this section we'll discuss how to set up ganks amongst other things.

Ganking:
This is possibly the most important part of job. Ganking provides map control, gold/kills, facilitates pushes and in a lot of cases can save an ally that would have otherwise met his/her end.
Once you reach level 6, recall, purchase any items you can to contribute towards your Madred's Bloodrazor and buy up to 2 sight wards.

Use these wards intelligently. You'll need to use them differently in every game. Usually, i place them just outside lanes prone to ganks or lanes that are struggling. Occasionally I throw one in the enemies jungle to track movement of the jungler and lane champions passing through.

Have Blood Scent turned off at all times. This gives your position away if trying to set a gank up. Only turn it on once engaging in combat.

To set up a gank, 'ping' the gank target and wait in bushes until your lane ally initiates. In some cases you'll need to initiate.
Common mistakes include your ally all of a sudden playing offensively despite being low on HP or your ally suddenly trying to push the lane. These are the exact opposite conditions under which you want to gank. Your target needs to ideally be closer to your turret than their own and your ally needs to feign a poor defensive tactic to allow the enemy to go in for a kill. Upon doing so, you should engage with Infinite Duress, immediately activate Hunters Call and Blood Scent and throw a Hungering Strike as soon as possible. If they escape with more than 15-20% health, don't waste time sitting in the lane. Return back to your jungle.

Jungling:
Yes, you should continue to jungle for as long as you can until your team starts regularly engaging in team fights. If your team is in team fights, ALWAYS help them before farming.
Whilst in bot lane, ideally after a gank, 'ping' the Dragon to indicate for both of the lane's champions to assist you in the kill. Do this as regularly as possible. It allows for a large amount of global gold + often scares your enemy team's mid into thinking he/she is about to get ganked, in some cases, this is exactly what you should do. Warwick is capable of killing the Dragon quite early on, however it is a slow process and with bad luck can result in a death or the need to abandon the Dragon. This always equates to lost time, and time is valuable. Try to jungle as much as you can, whilst also helping out lanes with ganks and pushes. The goal is to try and build as much of your core build as possible.

Your late game begins as soon as people start focusing solely on the Mid lane and abandon Top and Bot except when trying to go for a push whilst unnoticed.
This is where you begin really stepping it up. Depending on how far built you are, especially compared to the opposing team, you can really begin farming kills.
From here, always be aware of how best you can help your team before engaging in a team fight.

The Assassin:
It's quite common for an enemy champion or two to engage in a team fight with low health. Usually this is forced by your team, but occasionally, they are the ranged champions, who sit at the back anyway. Warwick is great for picking off these kills nice and fast. Engage with Infinite Duress, and activate all other abilities after the channel. This should lead to a quick clean kill. You should run immediately afterwards in most cases. If the enemy team is likely to focus you first, your best bet is running in when you see gaps in their defenses or a champion you can pick off.

The DPS:
With 147 AD at max level, 62 magic damage dealt per attack, 4% enemy health dealt as damage per attack and a near capped attack speed. You can really tear stuff up. With Hunters Call you max your speed at 2.5 attacks a second. Thats 522~ damage + 10% enemy health per second. That most definately hurts. Whether you play like this or not depends on how dominant your team is in that team fight. If they are playing definsively and you aren't going to get focused. Slowly pick them off by DPS'ing them down and score kills with Hungering Strike.

The Semi-Support?!?:
One aspect of Warwick which is too often neglected is his amazing ability to support his team in a team fight. With Randuin's Omen having a cooldown almost the same as your ultimate, this creates amazing synergy. But generally you provide a 40% attack speed buff to ALL allys, a 20% attack speed debuff to all enemy champions around you at ALL times, and you can for 3.5s (4 seconds if Wit's End is stacked), reduce the movement and attack speeds of all enemy champions by a whopping 35%.
The fact that these are massive team wide buffs really make Warwick worth having in team fights, even if you're low on health, it's worth staying behind allys to buff them at least with Hunters Call.

The Core Build:

Madred's Bloodrazor, sorceror's shoes, Wit's End and Malady
These 4 items MUST be in every warwick build.

Madred's Bloodrazor:
This is self explantory, the great aspect here is that it procs 5 times in 1.8 seconds when you use Infinite Duress. This is an instant 20% of the enemy's max hp.

Wit's End:
Attack speed + 42 magic damage per hit. This also procs 5 times when you use your ultimate. If you use the build recommended by this guide, when Wit's End fully stacks it's magic resist the active from Randuin's Omen is extended to 4 seconds.

Malady:
Cheap, solid attack speed + 20 magic damage per hit that procs 5 times upon the use of your ultimate. The magic resist reduction is amazing due to Hungering Strike + the damage from Madred's Bloodrazor, Wit's End and Malady all being magic damage.

Sorcerer's Shoes:
Magic resist as explained above. Warwick doesn't need Beserker's greaves as he become attack speed capped upon the use of hunter's call which has more than a 50% uptime anyway. Ionian Boots of Lucidity are wasted as he obtains enough CDR via Blue Buff and further amounts through his masteries + Randuin's Omen/ Frozen Heart.

Recommended Items:
Randuin's Omen:
This provides great survivability + a great way to turn team fights around. It can be built with heart of gold, allowing for some extra Gold per 5 whilst waiting to build this. Synergizes great with Warwick's pre-existing ways to support his team.

Frozen Heart:
This provides survivability to Warwick plus a great aura that really hurts carrys and semi-carrys. The extra mana means you can really sustain yourself that gap between losing your Blue Buff and re-obtaining it.

Other Items:
Frozen Mallet:
Pick this up if your enemy can often out run you despite your Blood Scent. This is basically a permanent Red Buff, but is often unnecessary.

Abyssal Mask:
Magic Resist plus a Magic Resist Reduction aura. This is a mage's job. But sometime's it's worth picking up if your team is AP heavy and their's is as well. Be aware of the wasted AP. Makes this item not so worth it's cost.

Zeke's Harbinger
Unneccesary Attack Speed, but provides a great aura to your team. If you have an AD heavy team, or snowballing AD carries such as Tryndamere who are being fed but wish to build other items, this item will greatly benefit your team. It also reduces the enemy's armor, this is often overlooked too. Be aware how great this syngerizes with a champion such as Tryndamere. They essentially recieve 40% attack speed from your Hunters Call as well as an additional 20% attack speed, 20 armor penetration and 20% life steal from this item.