AD ranged carry: Sivir
Hybrid ranged carry: Twisted Fate
Hybrid tank: Sion
AP tank: Vladimir
Jungle: Warwick

I chose Twisted Fate over Miss Fortune (who I usually use) as the other carry because his passive boosts the efficiency of the entire team (especially WW and Sivir). Also his Pick a Card and his Destiny are both such useful skills that (when used correctly/efficiently) are very disruptive to enemy teams.

Sivir should solo mid, the faster she levels the faster she wrecks turrets.

Twisted Fate should pair with Sion in bot lane; Sion's stun and TF's gold card can both be used on a greedy enemy for an early kill.

Vladimir should solo top lane, he's no easy gank for anyone. When waiting with Warwick at Golem, Smite the Golem before Warwick does. This will save lots of time and greatly increase Warwick's efficiency in the jungle.

Warwick should start with Blue Golem, but Red Lizard also works.
Protecting the jungler from the gank is crucial, Sivir and Sion (if red) or Vladimir (if blue) should accompany WW until minions spawn.

Mid Game (Ganking):
Warwick should be ready to be called into battle anytime after he hits level 6. His ulti's ability to suppress enemy champs while simultaneously replenishing his health makes ganking and focusing very easy.
Sion or Twisted Fate should slip into mid to stun for a kill if the opportunity presents itself; TF should be ready to teleport wherever he is needed from this point on.
Sivir should be a champ killing minion farming machine, and with TF's passive she should be very wealthy. She will be the primary target of ganks so she shouldn't wander off too far from allies.
Vladimir, just like Sivir, should be well developed and ready to help by this phase of the game. He should have a Rylai's which will make him much more useful in the coming team fight.

Late Game (Team Fights):
This team is clad with hard CC (Sion's stun, TF gold card, WW ulti) as well as "soft" CC (Vlad's pool, TF red card) with which to focus the enemy carry. Once their carry(s) are dealt with, the rest should fall with ease. With a well-built Sivir the push should be quick as her ulti raises her (and her allies) attack speed by a tremendous amount.