This is my first time making a guide.I have used this build in PvP and against AI.
I hope this build will help u out if you choose to play Caitlyn.
This build if played properly will give u a huge amount of damage late game.
Against AI (beginner) you can use your ultimate to immediately kill champs who have less than half their health left.(tanks are an exception)Also a good tank partner in PvP can help u to absorb damage while you deal massive damage to their team.Either Blitzcrank or Alistar make good partners with Caitlyn

Start of with Doran's blade for the life steal and extra damage.
Then get Berserker's Greaves for both the mobility which Caitlyn needs so badly and the fire rate increase beacause Caitlyn's base attack speed is quite low.
Move on to Last whisper for armour penetration and damage.
Followed up with Blood thrister for more life steal and extra damage.
Then finally move to Infinity edge for extra critical hit damage and extra damage.
After getting the second Infinity edge , sell the Doran's blade and get a third Infinity edge.
This third Infinity edge is optional.If you are doing badly or you are getting focused during team fights , banshee's veil may be needed. Black cleaver is also optional if your opponents stack armour.

Option 1:Go for all Attack damage Runes and Quintessence to maximise attack damage output.

Option 2:Change all the Marks of Strength into Marks of Might for higher damage late game but lower damage at the start.

Go for all attack damage masteries in the offense tree.
And take faster recall to escape ganks quickly. That extra one second may save your life.
The rest is for mana and 1 for extra buff duration. At level 14 you can go ahead and clear the entire jungle for all the money and buffs both red and blue.Get the dragon as well if you can.

Put your 1st skill into piltover to poke enemy champions during the 1st few skirmishes.And to clear the bushes of unwanted campers.The second skill yordle snare is vital for survival.Place snares in the bushes right at the edges to snare opponents who want to attack from the bush. (especially effective on melee champions)When placing snare you will also get the area around the snare revealed momentarily.The third skill 90 caliber net is your mobility skill.It can be used defensively to move back and slow opponents,as well as offensively to chase opponents.This skill can also allow you to get over low walls and jump out of enemy spawn from the high side to the low side.After putting one point in every skill,proceed to put 1 more in piltover followed by yordle snare.The second point in yordle snare is more for cooldown reduction than any other need.