This is my first build, on my favorite champ. Warwick is an absolute beast. I have been playing with him for the longest time, he was the first champ I bought. He isn't the fastest jungler, but he is safe and he is great for ganks. This build is mainly for being a dps carry after you jungle.

For runes, you will want armor pen quints, as most jungle minions have around 10 armor. The rest you will want attack speed, as it will help your passive which heals you for each hit. YOu will want for masteries improved smite and the neutral buff duration increase. Those are the most important, although the last utility also allows you to use more smites. For jungling, I usually start with a cloth armor and 3-5 health pots. The cloth armor will build into a Madred's Razors, then a BLoodrazor. Black Cleaver is great for the attack speed it gives, and the armor penetration. Bloodrazors deal 4% of the enemies' health every hit as magic damage, which will rape along with your ult by doing 20% damage for 5 hits. Bloodthirster is good for the AD and lifesteal. Wit's end will increase the damage of your hits by 42 magic damage, and is also proc'd by your ult. The MR stacks also help. Spirit visage will increase healing from your q, ult, passive, and bloodthirster, while also decreasing cooldowns, increasing health, and increasing MR. You can switch out spirit vissage for sunfire cape if the other team is more AD oriented. Also, your attack speed without hunters call will be 1.9-2, and with hunter's call near the max 2.5.

Make sure your team knows u are jungling. That means that one person will have to 1v2 lane top, so make sure they can handle it. Grab your cloth armor and 3-5 health pots and go to the two golems. This is my favorite starting place because with the 5% increased experience masteries, this will instantly giv you level 2, while taking a lot less time than blue. Also, when you take blue, people will leash for you, but sometimes they won't leave and they will take your xp. If you play with friends, blue buff may be a better starting point, but if you play pub, two golems may be better. Put your first point in q, and smite one golem to start it off. Use your q once or twice, and you should be at about 3/4 health. Upon leveling up, put the point in w. Go to wraiths, and use your q on a small wraith, and use your w. You can also pop a health pot. I find that going for the big wraith last helps me clear it faster. GO to wolves. Your smite should be up, but do not use it on the wolves. Do not even use your q on the wolves. Just take another health pot when the first one from the wraiths run out. you should level up, so put the point into your Q. Go to blue and smite the big golem, use hunter's call, use your q. take a health potion if needed. After you get blue buff, you can either go to the enemy jungle if they don't have a jungler, or you can go back to wraiths. When you get to the wraith camp they should spawn. Take wriaths while smite cools down, then take red, then golems. When you have 1050 gold, recall and buy Madred's razors and boots of swiftness. Razors will help you jungle faster, boots let you run away from / run towards ganks. At lvl six, start ganking by going to the solo top lane. Keep jungling and ganking. You also want to get dragon with bot lane around level 8. Try to get enough gold for a black cleaver. I do not suggest rushing bloodrazors too early as it does a percentage of health, but early-mid game you want black cleaver because enemies do not have a lot of health yet. Also try to pick up a vamp scepter, as this will allow you to stay in lane/ in jungle longer. Late game you will have completed your bloodrazors, bloodthirster, and hopefully wit's end. Then you can get sunfire or spirit visage. In team fights, your job is to catch running enemies, suppress an enemy carry with your ult, or just dps like mad. In team fights, I find Warwick will get cc'd a lot. That means that you shold not be the first to jump into a team fight. Wait for your tank to initiate and for some support/mages to lay down some cc, then ult in and start dps'ing. In 1v1 or 2v2, you will absolutely rape. You will ult, then q, then chase down with blod scent. Often, I find myself healing faster than the enemy can dps/ burst me. Also, instead of recalling, warwick can always just go to the jungle and heal up on minions.

Smite, for jungling, is great. At worst, it gives around 30 gold per 60 (that is if you aren't jungling). It speeds up your jungling by a lot, increases your survivability as your have to take less hits. It lets you take blue buff early.
Flash is just a great spell overall.
Ghost is ok, but you have blood scent. Flash is still better for escape, especially in the jungle where you can flash over a wall.
Heal is bad, clarity is bad, revive is terribad.
Ignite and exhaust are not as useful on Warwick because he doesn't need ignite as a finisher, he can usually just chase enemies down. He also doesn't need exhaust for the same reason.
Clairvoyance is not good. Even if the other team has a jungler, let you support take cv if you really want them to check a minion camp.
Teleport is not useful, b/c you have nothing to tp onto in the jungle. For ganks, blood scent should let you get there fast enough, especially at later levels.
I have heard that rally helps jungling, but I would rather have flash still as warwick has no good way to escape.