Before a game, I decided to try a full attack speed build for my chosen champion. And that champ was Ashe. I was buying attack speed items as I was playing. In the end, Ashe was shooting arrows so fast it was funny, yet effective. (Highest I managed in a 30~40 min game was 2.500.)

Courtesy

You can start with either Doran's Blade or Boots Of Speed

Boots Of Swiftness is the recommended boots, but Berserker's Greaves is only 1 movement speed less, and is the only one that buffs attack speed

If you are earning more gold than expected, build Phantom Dancer first

If you are still earning more gold than expected, build another Phantom Dancer (2 at most)

The other option is to build The Black Cleaver

Be weary in choosing whether to buy a Dagger or Recurve Bow, as some builds only require one and not the other

Scapegoat: just buy attack speed items

In serious game:

Level up as according to gameplay style/what is required

i.e. This may mean putting one point into Hawkshot [E] in early game

In non-serious game:

Level up Volley [W]

Level up Frost Shot [Q]

Level up Enchanted Crystal Arrow [R]

Level up Hawkshot [E]

Pros:

Buffs attack speed

Buffs critical strike rate

Buffs cooldown reduction eventually

Will become so much more easier to chase down enemy champions, and finish them off

Ashe looks funny whilst dealing non-spell arrows at such a high attack speed

Cons:

May have to play defensively early game whilst farming

Can still be squishy in mid/late game stages

Expensive build