Im not gonna bother you with tips like pay attention to the lanes. With warwick you have a built in way to do this. When the W passive is up just look at whats going on and most of the time you can walk across half the map and still get to the target in time to get a kill.
My main tip here is to try to invade the enemies you can, and steal some camps while maintaining frequent ganks.

For ganks you can abuse E to tower dive if the enemy is under 20 % hp otherwise dont do it unless maybe youre risking it for first blood. But even then its not worth.
Other than towerdives your standard gank is whenever enemy is near or under half hp you can just run up to them from some side and most of the time turn on E before you get to them to fear them faster. If they are strong maybe save E to soak up some dmg.
If you have Ulty then definitly save E and try to hit them a few times first, then Q and E and then hit some more and ulty if they are getting away.
If your laner is healthy this is all easy if he is low or died already it is a bit harder but you get used to it pretty fast.

Important thing to remember with warwick is that he can heal a lot when he is low. so if there is a fight in the jungle or the river and you get droped to next to no health you should try to just get out of there and run to the closest jungle camp. Many times you can get out of a fight with 100 hp spend 5 seconds on a camp nearby and come back to the fight full hp to get a triple kill.
You should keep this in mind and make sure you dont INT. just because you started a fight doesnt mean you have to stay in it untill you die. get out do some laps and come back in. your speed is amazing and zou can do alot more by making the enemies chase you than by standing still untill you die.

Many times people will chase you while you are low you will get to the first jungle camp Q that and then just E and get back to killing the unsuspecting squishy that is chasing you. If you manage to get them to chase you to some ally with a stun or a heal/shield you can definitly win the fight.

What I like to do is I take rebuff then blue and gromp if im botside otherwise blue gromp, red on topside.
You take 1 lvl Q W E and you go straight to enemy camp. blue camp if ure topside. redcamp if ure botside. Plant the ward in their bush and wait for them to start doing it.When they get it to 600 hp smite it and insta W E and start autoatacking.Dont forget to W if they are over half hp, and you dont need to do it if they are low hp so its already active. Dont rush with the q as you can most of the time use it after second autoatack and hold it and it makes you follow their flash or escape while they are panicking because they did not expect this.
If you get their second buff and a kill then take all the camps on this side of enemy jungle and proceed to gank whoever is lowest arround you.
If you just steal the buff but they escape just get out of there. They are already 200 gold and a lot of xp behind and you ruined their pathing it is very bad to die now to a midlaner coming to help them.

Most of the games I do this and usualy I get the buff and a kill and then I gank toplane and get another kill. Then I do scuttle and wolfs and recall for a tiamat already. After recall you want to try to rush the second scuttle first and then do raptors and golems and gank mid or bot.
If you did all this now you could already have 3-4 kills and its time to hur*** the enemy jungler further. take his raptors and his gromps whenever it is possible.
DO NOT GREED. taking the raptors and gromps is great and easy if you know he is on the other side of the map. if you have no vision on enemy jungler and enemy botlane and you are taking gromp YOU WILL DIE 90% of the time. if you are taking raptors or golems and you dont know where enemy jungler is + top or mid you WILL DIE 90 % of the time. sometimes you will save yourself by ultying away but that is 1 kill less for you that u wasted ulty to run away.
Also the ulty is a bit tricky because sometimes you hit people you are trying to ulty away from so then you die and you lost ulty which is super bad.


If you ****ed up any of the steps above and find your self in a situation where you gave first blood instead of taking it then you need to forget everithing and farm to lvl 6. exceptionaly if you still have flash you can gank someone that your passive marks and is under 30 % hp but most likely you should just farm to lvl 6.
Once you have lvl 6 tiamat and sheen you can go back to killing anyone your passive has marked and try to snowball back into the game.

If that fails for some reason you die again and you are even more behind then you need to make a serious effort to take drakes and heralds with herald being your priority. If you get both heralds you can push 1 lane to inhibitor and win some unwinnable games. Or at least create enough pressure to let your toplaner roam while you farm the jungle to get back in the game.

When you have compleated Hydra and Trinity you become a beast. You can deal insane dmg and take a lot too. After you finish deaths dance as well you can 2v5 pretty much if you have someoene with you with some cc or dmg.
If the enemy team has 4-5 champions that can slow snare stun or otherwise peeel you might want to think about getting a completely different jungler. But if you are already locked in then you need to make sure that if you are fighting it is really 5v5 and not 4v5 or 3v5 because you wont be able to reach anyone.
Ideally in a 5v5 scenario your team has a lot of cc as well and some extra targets for the enemy so you can run arround and get an angle to jump on their adc or anyone who is fed.

Even better if they have tons of cc you want to apply pressure to outer lanes and make the enemies spread out. Pressure means that you are pushing top when your bot is pushing and pushing bot when your top is pushing. If you are pushing top you should do it with toplaner and not alone and same on bot.
If you find yourself in a situation where you are pushing top and your whole team is in the base/ jungle/ defending towers you can expect that you will be dead in under 30 seconds.