First I want to Apologize because english is not my native language so if you see any mistake please just ask me to fix them.
Ashe is a really nice carry and a heavy famer. I am going to show you how to play ashe on a REALLY carry way (damage + attack speed + armor penetration + critical chance + life steal) and explain you my differents choises.
Please leave some comment to help me to make my build better.

One of the perennial favorites of summoners in the League of Legends is the Freljordian beauty known as Ashe. She is a direct descendant of Avarosa - one of three legendary sisters who each claimed dominion over the scattered tribes dwelling in the icy tundra of northern Valoran. Ashe mirrors her ancestor's unparalleled mastery of the bow, earning her the title of ''The Frost Archer'' the way Avarosa did during her era. Ashe is a literal Princess amongst her people, though she prefers to be addressed by outsiders as her tribal title rather than any form of royal moniker. There are those in Freljord, however, that would prefer to address Ashe as the ''late'' Frost Archer; the other two tribes that are descendants from the Three Sisters are historically sworn enemies of Ashe and her people. Having survived more than one assassination attempt in her life, Ashe is always aware of her surroundings regardless of where she is.

Ashe originally came to the Institute of War in service of the League summoners, seeking enough influence and favor to finally bring peace to her realm - one that has endured civil strife since Avarosa and the Time of the Three Sisters. With countless victories under her belt, she has started to put the influence she has earned in the League to use. Rumors abound that Ashe has begun to associate herself with fellow champion Tryndamere outside of the Fields of Justice. While she denies such talk as frivolous, all eyes will remain on Ashe now that her success in the League may finally allow her to restore lasting peace to her people.

''It's a good idea to spread out. Ashe can hit five birds with one arrow.'' - Tryndamere

- Greater Mark of DesolationX9 This rune allowed you to make strong damage even if the ennemy have some armor. That is a nice choise if you play versus some kind of tank you will nt make low damage.

- Greater Quintessence of Desolation X3 for the same reason as the Greater Mark of Desolation
- Greater Seal of Resilience X9. You will see that ashe is a really nice carry but you don't have a lot of health if you don't take any banshee veil. This rune allowed you to "tank" a little bit more the ennemi damage. That is nice versus the ennemi tank because they don't make a lot of damage.

- Greater Glyph of Warding X9. That will help you to win lane versus other carry

I am going to explain my differents choise on my stuff

- First, I take the doran's blade. You take this Item because in early game you will need some extra damage. This stuff is used for farming or for early agression if you play at mid or versus a solo top. You also can take boots of speed + health potion X3 but you will not have enough damage to kill your ennemy.

- After that, I take a Berserker's Greaves this permit you a high attack speed and a nice movement speed. You also can take Boots of Swiftness but will not make as damage as with the Berserker's Greaves but that's a nice Item combine with the frost shot of ashe.

- Then, I take a Vampiric Scepter to be allowed to farm longer on my line. You also can take the zeal before but I prefee taking the Vampiric Scepter as fast as possible.

- Now, I take the phantom dancer . That help you to move and attack faster and make for critical damage but you will have low damage that's why you can take a B.F Sword to finish your bloodthirster or even take an Infinity Edge .

- I finish my bloodthirster To have some more damage and life steal.

- At that moment, I take a last whisper to have some armor penetration and and attack damage. You can also take a Black Cleaver but I prefer taking it after.

- Now, you must take the Black Cleaver because that give you some attack damage/attack damage speed and his passive give you some armor reduction. That's a quite nice item for ashe.

-Focus is ashe passive skill. That give you some % critical damage. That is really nie in early game at mid to just make a shot on the ennemy hero. That's really nice with teleportation too because you can make really effective gank but be sure that you are not waisting this skill on minions.

- Frost Shot. This a rzally powerfull slow. If you have a nice atack speed you would kill AD annemy really easily. But be carefull this skill cost a lot of mana so be sure to don't activate it versus minions or taking the blue buff.

- Volley. This the skill tou will use the most because you can make nice damage at a nice distance. That can also slow ennemy and hit some minions at the same time.

- Hawkshot can help you to give some extra gold. This skill is also really usefull before attacking a tower to have the vision to see if somebody is ganking you but that is not the most usefull skill.

- Enchanted Crystal Arrow is Ashe's ulti. With it, you can make some nice damage and stun your target for some time. You MUST use this skill to kill the ennemy or sometimes to finish an ennemy who is recalling under a tower.

I really advice you to take the same masteries as me because that give as damage as possible but if you have any suggestion pease write a comment to help me to make my build better.

-flash : this summoner spell will help you to save your life or even to finish a kill

- teleport : I really advice you to take this sommoner spell because you will be able to help your ally or recalling in early game without deleated your level.

- heal would save your life and winning a line versus some other carry.

Pros :

- Heavy damage
- Good farmer
- Good at killing tower
- With this build you can jungle really fast in late game.

Cons:

- Low health if you would be focussing in team fight you will died really fast
- Low mana

Ashe is not a really good jungler so if there is some jungler in our team they will take the jungle but if there is not any, with this build in late game you would jungle eazily and fast but be careful to don't fight versus ennmy jungler during the time that you are jungling.

Ashe is a carry so in team fight you must becareful to don't be foccussing. Before the engagement you would make some volley to make the ennemy health low and use. During the team fight if the ennemy team is backing you would use your frost shot on the ennemy carry or your ulti to take some free kill. If your team is backing you would save some ally with your slow or with your stun.