Caitlyn is a character everyone wants to get in ARAM. She is fairly easy to play and does a ton of damage. Her poke has great range in Piltover Peacemaker. She also has a great getaway in 90 Caliber Net. This allows you to poke and get our of danger immediately. Reducing the damage you take and prolonging your harass.
In the beginning of the game, keep poking the other team using Piltover Peacemaker, lay traps in crucial parts of the bush (the areas where people enter and your team can immediately go for a quick kill). Use 90 Caliber Net to getaway. Once you get items and level up, eventually you can lay traps everywhere using it as another source of damage.
Try to spend as much as as possible in the bushes as Caitlyn gets a head shot every number of shots taken while in the bushes. Do not kill yourself trying to stay in the bush when its not safe though.
Caitlyn relies heavily on her auto-attacks and you should start by building Infinity Edge. Follow that with attack speed boots, life steal/attack speed. If you are feeling lazy get Hurricane earlier and just farm. You can potentially get a lot of kills with Hurricane early as people are usually bunched up. Always attack whenever anyone on your team engages. Every shot you can take on a champion counts. Just be ready to back off with 90 Caliber Net. If you are facing champions that can teleport and target you, before you engage lay a trap where you are going to stand to attack, that way as soon as they teleport and get trapped, you can 90 Caliber Net out and continue to attack safely. Make sure you are constantly putting pressure on the other team. As Caitlyn, your teammates will be expecting a lot from you.
Use your ultimate to finish off opponents rather than causing initial damage. If the battles go accordingly, you should be able to deal the bulk of your damage in the engage. They would usually run in this case and this is where you go for the final kill. It seems like a waste otherwise.