Ashe, played unconventionally with its starting items.

ArP-Standard AD Carry
Dodge-Always helps
Magic Resist-Nothing else here really helps

21/9/0 Standard AD Carry, Dodge in defense for accommodate runes

Health Potion x10: Massive lane sustainability, allows you to stay longer than a Doran's blade
Doran's Blade x2: Damage, survivability comes from these
Berserker's Shoes: Attack speed helps
Zeal: Compensates for lack of movement speed from no Boots of Swiftness, also crit/attack speed
Cloak of Agility (Used in PD): Crit always helps
Phantom Dancer: movement speed always helps ashe, crit and attack speed too
Cloak of Agility (Used in IE): Crit always helps
B.F. Sword: +50 AD and makes you more intimidating
Infinity Edge: Your crit hurts this much more and you're a walking machine of pain
B.F. Sword (Black Cleaver): More AD is all you need now
Black Cleaver: Used on heavy armor teams, helps your ArP in general

Sell Doran's as needed

Bloodthirsters x2: Here is a choice point, whether you're needing lifesteal or more AD it's your choice, but in the end the two bloodthirsters make you unstoppable.

Note: Black cleaver or a bloodthirster may be replaced with a Quicksilver sash or a banshee's veil if you feel as if you're much too squishy.

Final Items: Berserker Shoes, Infinity Edge, Phantom Dancer, Two Bloodthirsters*, Black Cleaver
*: Or banshee veil/quicksilver sash in place of one

Arrow > Volley > Frost Shot > Hawkshot

I thought about starting with Hawkshot at level 1 for extra gold, but it's honestly up to you

Flash: Positioning, prevents bad messups
Ghost: Good for escape

Use both, they help you chase or prevent you from dying

Always be doing this, always be killing something - it's necessary for you to last hit everything so that you can get those items as quickly as possible. Lane pushing is also a necessity as towers make your wallet happy.