Use Cait's edge on survival (longest range among ADC champs and trap & net skills) to your maximum advantage, but don't be overconfident. Farm carefully in the beginning until you get Berserker's Grieves and Vampiric Scepter. If you don't farm well it's very difficult to prevail against champs that become dramatically stronger in late game (e.g. Vayne). Make sure you place the trap where you would place a ward. (Not only in bot, but later on in your/Enemy's jungle as well.) You may save yourself AND your teammates when you're running away from pursuers.

Cait can work with most supporters, but tankier ones are preferred. My favourite picks are Ali and Blitz. Soraka may be helpful in maintaining a strong position in the early game and farming in peace, but it will be difficult to initiate a fight. Janna is also good for providing movement speed+ and throwing enemies airborne.

While many perceive Doran as Cait's must-have starter, I think it's better to obtain Vampiric Scepter as soon as possible. If you're with a healer Doran will be fine too.

BUT, if low on cash, Berserker's Grieves should be obtained PRIOR to VS.

I'd like to go Bloodthirster instead of Infinite Edge because it better ensures Cait's survival with life steal. The critical damage will come from Phantom Dancer. I usually get Last Whisper first before completing PD (I will get Zeal first however) for armour penetration.
If your early performance was very strong, complete VS and then get Infinite Edge.