Hello and welcom to Caitlyn - Sheriff of Piltover
Lets begin with Skills:
Piltover Peacemaker- Skillshot that does flat damage + 100% AD physical dmg, has a long range and is extremely useful for farming minions, getting those runners, and just firing into the fog when a low hp enemy is running :)

Yordle Snap Traps- Traps that are set by Cait, immobilizes enemies and deals damage to the enemy for 1.5 seconds, has a small animation set before it can be activated so it might be more useful to cast .90 Cal net first or place it a few steps ahead of your champ, try to time it where it is activated as soon as you step on it for ultimate pwnage.

.90 Caliber Net - Very useful escaping/catching enemy champs, as well as a good source of damage, doesn't scale with AP very well and should be used sparingly considering the mana cost.

Ace in the Hole - Cait's ultimate, an awesome skill that does tons of damage (scales to all BONUS ATTACK DAMAGE 2x, so lets say you have a bf sword(50) and a pick axe(25), it would scale to +150 additional damage on the ultimate, pretty nice if i do say so myself. Love using this skill with Yordle traps, they get revealed and try to recall or get a buff, pop my ult and once it starts to channel, it is impossible to escape the bullet.


Jungling: Can be done with Caitlyn, if you feel like jungling then fine but this build is not for it.
Lanes:
Caitlyn excels at top and bottom lanes in 5v5, not so much middle since there isn't much brush for her to use for her passive, but her extreme range makes up for it with easy harassing on the enemy champ. You should be using your passive doing all the time while laning and when you can in team fights. Playing in 3v3's try to take the solo top lane and in 5's leave the mid for some one more capable like karthus or ashe.

Runes:
I chose mp5/lvl seals, ArP Quints, ArP runes and runes, they all seem to work well when i play caitlyn, ARP runes to hit harder, mp5/lvl runes for laning longer,

Masteries:
Used a typical 21/9/0 for masteries, could use masteries however u like, i just like this set up. Ranged DPS so speccing in offense shouldn't even be questioned

Items:
As for items, i messed around a little bit in normal games and i found this item sequence to benefit me the most, doran and health pot to lane, i found dorans blade to give all around good stats for caitlyn, following up with Zerker grieves, and a dagger if you can get it. Since caitlyn has incredible range it is extremely easy to harass with your auto-attack. Once you can afford Malady head back asap, awesome with Caitlyns range and passive, constantly stacking damage and -MR helps. Blood thirster is what you should get next, following up with Zeal, building into Phantom dancer,then build it into a Infinity Edge, and if the game is still going go ahead and follow that up with another Phantom Dancer, crit chance is unreal and can solo minion waves like nothing. Usually games dont last until the 2nd Phantom but if they do once u get that maxed out it should be no problem ripping away at squishies, i love seeing people get crushed under the weight of dmg, and since her range is long, enemy champs are still subject to harass and abuse while turret hugging, and its always funny picking up 2 kills while they thought they were safe by their own turret ahahahahaha.

Skill Sequence:
i have tried starting with .90 cal net and snap trap, but piltover peacemaker is what i have found gave Caitlyn most first bloods, as enemies try to run and at lvl 1 most just group up.
Get your ulti asap and head to either top or bottom lanes for a gank, after a little rabble rousing with the enemy ult should be sufficient enough to take down one person. Just remember to use yordle snap traps and your ulti in combination, can give you some easy kills.

Summoner Spells:
I chose ignite and ghost, i found it works best. Ghost can be used in combo with .90 cal net very easy to escape from some champs but just remember you cannot go over all walls with .90 cal net so be careful where you choose to do it at. Ignite i grabbed because a few games i played i tried flash + clarity, but even though i had enough mana for farming, late game it ended up doing no good at all and there were so many kills that got away from my grasp because they had 5-50 hp.
Ignite: Pick up those pesky champs that get away from you with little hp
Ghost: Always handy if u need to escape from a tight spot or being chased with little hp, saved my *** plenty of times

Team Battles:
Once you start engaging in team battles, always stay on the outside and let the enemy get nice and comfy with the tank, you should use your R when you think you should and never in team fights unless you have a straight line at the target, since your ulti can be blocked by any enemy champ, its imperative that you make sure your ult hits its target. Dont be afraid to spam your E, it does a nice ammount of dmg and can change the tides of team battles drastically. Start off with targeting the squishiest first, should go down like nothing once you have black cleaver, watch as their hp just drops and drops and drops, including the passive, caitlyn does good dmg and can reduce even the strongest champ useless.