This, is Critwick. Critical Strike Warwick. A lot of people say that Warwick has fallen out of favor, or that he just isn't very good. I am here to disagree, and to further say, that the majority of the League of Legends community has been playing Warwick WRONG. (Ok, maybe not, but drama is fun, so...)
Regardless, I want to quickly get out of the way that this is NOT a troll build. This actually works. Now, I'm not saying that it's really broken, or that it should be nerfed, but this does catch a lot of people off guard, and it can be a very effective way to win games.

I tried this build on complete accident, and it worked absolute WONDERS for me. I was messing around with stupid junglers with a friend of mine, and I decided to go play a "Full Crit Warwick."
With many laughs and easy clears, I got through the first 7 or so levels, with a few ganks and lots of farming. Not too much to note, for our build didn't diverge from the typical Warwick until a bit later. But, after I had my Berserker's Greaves, Stalker's Blade - Warrior, and my Phantom Dancer, my damage, and potential, during ganks and teamfights, finally began to show. With free sustain from Eternal Thirst and Hungering Strike, insane movement speed with Blood Scent and Phantom Dancer, and the raw damage from his autoattacks with crit, in ADDITION to the insane base damage on Hungering Strike all together makes Critwick a huge threat to the enemy team.
By the end of my first game with Critwick, I ended 24/9/12, in a very close victory with the overall score of 58/53, my team emerging with the victory.

A pretty basic rune page.
We take the five Greater Mark of Critical Chance to get a couple early crits for laughs; that ~5% could save you, but don't count on it too much.
The rest of our marks we use Greater Mark of Attack Speed, because attack speed is... Well... It's just good, ok?
For seals, we take full Greater Seal of Armor because without them, clearing the jungle, especially early, would be extremely hard. A bit of extra armor never hurt anyone.
For our glyphs, I advise 9 Greater Glyph of Scaling Magic Resist because when mages start to get damage later, you'll be glad you came with a couple extra tank stats. I don't see any strong arguments to use anything else. We don't need cooldown reduction, and flat magic resist in the jungle doesn't do much of anything for us, because jungle monsters do only physical damage, and we shouldn't be ganking too much early, except when our Infinite Duress is up.
Finally, for our quints, go with 3 Greater Quintessence of Attack Speed, because attack speed is great.

Now, I know that some of you are wondering why attack speed instead of attack damage in our rune setup. This is because since we are going with a crit build, the more autoattacks you get off, the more crits you are going to get. Also, since Warwick heals off of every autoattack due to his passive, Eternal Thirst, more attack speed means more healing. PLUS, our first bit of attack speed in an item will be from our Berserker's Greaves, because we are going Stalker's Blade - Warrior, so until that point, we would have painfully slow attack speed when our Hunters Call is off cooldown.

Now for Masteries: Not much to say here. Pretty standard AD Melee page, 18/12/0, taking Warlord's Bloodlust as our Keystone Mastery, because we are playing a crit champ.

So, I don't think I need to explain Smite... You need it if you're jungling. Yes, even for Warwick. And Critwick. And every other jungler. Ever. Period. Smite OP.

Now, Flash is a bit more debatable. Some people prefer Ghost, which is fine with me, but I still believe that Flash is better.
Here's why:
Flash is more unpredictable, and allows you to surprise your enemies with a Flash into Infinite Duress for a gank, or to make a pick in a teamfight. You can also flash the walls on dragon and baron pit for a steal, flash over walls for a chase or escape, dodge important skillshots, like Ashe's Enchanted Crystal Arrow.
With Ghost, sure, you can go super fast, but we don't need all that movement speed when we have Blood Scent, Phantom Dancer, Berserker's Greaves - Alacrity, AND Statikk Shiv's bonus movement speed. The short range teleport with Flash is much better, simply because it can do more, AND we don't need the overkill speed that Ghost provides us.
Regardless, if you still prefer Ghost, it is viable.
You really don't need anything else, such as Teleport, Exhaust, and Heal. They are all not necessary.

Start with W, Hunters Call, at level 1. This will give you, AND whoever is giving you a leash, a temporary attack speed buff, which will allow you to kill that first camp super fast. After that, get Q, Hungering Strike, for the sustain and good damage.
After this, keep leveling Hunters Call and Hungering Strike until you're ready to gank, be that level 3, 4, 5, or 6. If there is an early kill to be had by your level 3, get your Blood Scent and pick up the kill. Same thing at level 4 and 5, with those levels respectively. If there are still no easy kills at level 5, still get Blood Scent now. This is so at level 6, when you get your ultimate, Infinite Duress, you already have Blood Scent to execute the gank with efficiency.
After this, max Hunters Call, followed by Hungering Strike, and then Blood Scent. Put points into Infinite Duress whenever you can.
NOW, WHY do we max Hunters Call over Hungering Strike? The same reason we get attack speed runes over attack damage ones. Since we are playing Critwick, we will spend the majority of our time autoattacking, not casting spells, so attack speed is very beneficial. Also, like I said before, the more auto attacks you get off, the more crits you get, which is what we want for this build: as many crits as we can get.

For all of the newer junglers out there, here's a basic idea of what to do, when to do it, how to clear, and such things.

Lets go over our first clear. Start with a Hunter's Machete and 3 Health Potions. This will keep you very healthy. Start in your bot side jungle: For Blue Side, this would be the Ancient Krugs, or for Red, Gromp. After this, do your buff camp, be it Red Buff, if you started Ancient Krugs, or Blue Buff, if you started Gromp. After this, either go kill Scuttle Crab to get health back, then do your other buff, or go straight to your other buff. You may want to back ( Recall) after this, gank, or continue to clear your jungle camps. The choice is yours.

For Ganks, you typically want to come through the river bush, regardless of the lane, ult your target, then follow up the rest with Q's and autoattacks. Please keep in mind that if you burn your target's Flash, that IS a successful gank. That flash isn't gonna be up for another 5 minutes, and in that time, your laner could kill them, you could come in for another gank and kill them yourself when you ult is up again, and it will take away from their kill pressure in the lane without the surprise element that Flash provides.

As a jungler it is YOUR JOB to secure Dragon, Baron Nashor, and your buffs. DO NOT try and take either Dragon or Baron Nashor without smite. 9/10 times, it will be stolen. Unless there is NO pressure on the objective from the enemies because you know that none of them are near, or that they're dead, ALWAYS smite them. This may seem obvious, but it is ESSENTIAL for a jungler to be able to smite these objectives and keep them for their team.

Remember, as Warwick, you are not an early ganking machine. Your focus should be to get gold as fast as possible, for your items. This is usually achieved best through power farming when your ult is down, and ganking a lane when it's up. Also, if you see an empty lane, do not be afraid to go and tax (go into another lane and take CS or sit back and collect free experience) it. You will level up and earn gold much faster than you would if you were in the jungle. This effect is amplified nearly threefold in mid lane.

Now to the meat of this guide. THIS is what makes Critwick who he is.
First of all, lets talk about jungle items.
I prefer Stalker's Blade because it's my favorite. I suppose Skirmisher's Sabre and Tracker's Knife are just as viable, so use whichever you prefer.
Now, regardless of which tier two jungle item you got, Enchantment: Warrior is essential. It gives us our first bit of attack damage, which will allow us to build an attack speed item next, giving us our first bit of crit earlier.
The reason we are not going Enchantment: Devourer is because we aren't going for the traditional on-hit build for Warwick with Blade of the Ruined King and Frozen Mallet and stuff like that. This is Critwick, so we don't need that on-hit stuff. Just raw damage, which is what Enchantment: Warrior provides.

Now, getting into the actual crit. We get three crit items, which will get us to 80% crit chance, which is enough for us, considering how at full build with our Hunters Call we will have reached the attack speed cap.
The three crit items we get are: Infinity Edge, Statikk Shiv, and Phantom Dancer.
In our build, we will always want to get either Phantom Dancer or Statikk Shiv right after our boots. You can get either one now, but just keep in mind that you won't get the other one until your very last item.
After either Phantom Dancer or Statikk Shiv, get an Infinity Edge. This will get you to a minimum of 50% crit, regardless of your runes or which of the previous two items you got first. Infinity Edge will give your crits a bit more "oomph."

At this point in the game, you may begin to die a bit more, because you don't have enough health to stay in fights for a long time, and your Eternal Thirst and Hungering Strike just aren't cutting it. This is why we are now getting our lifesteal item. I recommend either Bloodthirster, for the shield and massive 20% lifesteal, in addition to the crazy high flat AD of 75. Another option would be the Blade of the Ruined King, for the percent max health shred against tanks, in addition to the active slow. (Quick tip: When using Blade of the Ruined King, use the active BEFORE your Chilling Smite, because Blade of the Ruined King steals more movement speed, for longer. Regardless, I would not use them together, as you get a lower movement speed boost than you would if you used them separately.) Another option, if your team lacks waveclear, or you want to splitpush, is Ravenous Hydra. I haven't done too much testing with this item on Critwick, but it seems like a fair option.

Round off the build with either the Phantom Dancer or Statikk Shiv: whichever you didn't get earlier, Enchantment: Alacrity, and Elixir of Wrath for even more stupid sustain.

I'll make this simple: You are now the ADC. Granted, you are melee, but you do as much damage as one, so treat yourself like one. On the other hand, keep in mind you will have crazy sustain at full build, so don't be afraid to go in with your front line. Your sustain is surprising, so, in most 1v1's (except, like, Tryndamere, and maybe Olaf), fight it out. You will most likely win, due to your insane damage and sustain.

Basically: Kill everything.

Finally! After reading this ancient text of how to play the forbidden Critwick, you have all the tools to go win some easy games! All other things aside, thank you so much for reading this guide on a funny little build that I have come to call, "Critwick." Please tell me how you do with this build, and if it was fun or not. Any and all comments are welcome and will be read, serious or silly. Constructive criticism is very much appreciated, and new ideas will be tested and possibly added, with credit given.

Best of Luck to all of you!
-Trininja