When you are team fighting, you're mission is to wipe out their backline and tanking all their damage which you heal almost fully with your Q, and if you're almost dead just zhonya's and let you're team handle the rest for you.
Usually the buttons you press are in the following order:-
R->Q->W->Hextech Gunblade->Q->Zhonya's->Q , BUT BETWEEN EACH SEQUENCE TANK SOME DAMAGE

I've tried that build before, sure you get alot of attack speed but its just that the autoattacks don't do enough damage with an AP Warwick build, because only 50% of your AP is added as damage to your base autoattack damage, but with Guinsoo's and Rageblade you get some AD for you to get at least some damage on both your autoattacks and ultimate.
The reason I like Hextech Gunblade is because of its Unique Passive which has synergy with his Q and as a bonus you get an active that can also heal you when your Q is on cooldown, all the while its active does alot of damage to the enemy because of its AP ratio.
The reason I like Guinsoo's Rageblade is because of of it's passive as well, which causes it to trigger on-hit effects an additional time every other attack, just like a ****tier sated devourer because you have to stack it six times first then the passive will take effect, but that is no problem for Warwick because of his Ultimate which stacks it 5 times all the whilst it applies other on-hit effects, and with another autoattack it's fully stacked, causing it to apply 3 on-hit effect every two autoattacks. This gooes well with the on-hit effects of Nashor's Tooth, Wit's End and Guinsoo's which add up to +70 magic damage per autoattack, thus autoattacking twice inflicts +210 magic damage onto the enemy causing even more healing to Warwick because of his Passive and the Passive of the Hextech Gunblade. This causes him to be a healing machine.
The reason I don't like the AP build, is because it depends too much on the healing off his Q, which doesn't come off cool down enough to be a consistent source of damage. Not only that but this build only gives him 40& cooldown reduction at lvl 18, whilst my build gives him 45% cooldown reduction causing his Q to come up more often.
To top it off I gave him a Cinderhulk enchantment, to provide consistent damage to openents and consistent healing when he is dueling with enemy champions. Add in the increased healing of Spirit Visage, he becomes a dueler no body can win against thus can split push like a beast late game.
The only problem with him is that he sometimes over heals and can be easily bursted down thus the reason I gave him alot of Health runes and finished the build with Zhonya's and Spirit Visage to give him a chance to not be bursted down.