I usually play Warwick as a DPS and gank when a lane needs it. Once you get your Ultimate, Warwick is good at jumping other champions and helping push a lane.
Eternal Thirst (Passive) - Each autoattack deals XX additional magic damage and heal him for the same amount. Each successive attack against the same target will stack this amount of healing and damage up to a maximum of 3 stacks. The stacks remain for 4 seconds.
Hungering Strike - Strikes an enemy for the higher value between the flat magic damage or a percentage of the target's maximum health, and heals Warwick for 80% of the damage dealt. Minions and monsters can only be dealt the flat magic damage.
Hunters Call - lets out a howl that inspires all nearby allied champions, increasing Warwick's attack speed and all affected friendly champions' attack speed by half of the amount for 10 seconds.
Blood Scent (Toggle) - senses enemy champions under 50% life within a certain distance of him, revealing them until they heal, leave the area or Blood Scent is deactivated. While sensing an enemy, he gains extra movement speed.
Infinite Duress (Ultimate) - lunges at an enemy champion, suppressing the target for 1.8 seconds and striking for magic damage five times in 0.334-second intervals. Warwick gains 30% life steal for the duration.
For Items I buy the recommended items starting with Doran's Sword then working towards Madred's Bloodrazor because the items passive allows basic attacks deal bonus Magic Damage equal to 4% of the target's maximum health then continueing to buy the rest of the recommended items.
In longer games after purchasing all recommended items I usually sell Doran's Sword and purchase items that increase critical strike or lifesteal.
My leveling up pattern usually is as follows:
1 - Hungering Strike
2 - Hunters Call
3 - Blood Scent
4 to 5 your choice
6 - Infinite Duress
7 to 10 your choice
11 - Infinite Duress
12 to 15 your choice
16 - Infinite Duress
17 and 18 your choice
Usually I max Hungering Strike first then Hunters Call leaving Blood Scent for last.