Ashe is the example of an Attack Damage Carry that is super squishy. In order to keep them at bay, people build some tank on her, however, increasing her attack speed and giving her life steal would make her able to 1v1 any champion in the game. The order I placed upon is one of many but it is the most used build and also the most effective. Ashe range makes her able to keep melee fighters at bay but does not outrange other ranged attackers such as Caitlyn or Varus. Her abilities make any champion slowed and gives her time to hit them before they approach her. Her Q gives her basic attack slowing affect. Her W is volley which shoots numerous ice arrows in a cone outward direction, damaging and slowing any target hit by the arrows. Her E is Hawk eye which allows her to reveal an area of the map if shot in that direction, however, the passive of her E also gives her bonus gold for every minion and monster kill. Her R/Ultimate is the Ice arrow which travels the entire map until it hits a champion or goes off the map. Her Ultimate also stuns the champion it hits. Her W scales off attack damage but her R scales off of Ability power so the ultimate should be for the finish of an out-of-reach champion or to initiate.
You would like to chain her abilities this way: E, Q, AUTO, R, AUTO.
Her E is completely useless in a fight and should always be maxed last.

Leona is not an AP carry but if built right, she can do some serious damage and not be harmed.
Leona is built mostly as support due to her natural tankiness and two stuns that she has. She usually initiates and has CC on her ult. However, if she is build AP, her abilities become more effective and will most likely ensure a kill every time against the ADC. Leona has two stuns and she can use them to initiate. Her Q is Shield of Daybreak which is a melee stun. Her W is eclipse which shields Leona; giving her bonus armor and magic resist. After a certain amount of time, the shell explodes and deals magic damage to surrounding enemies. Her E is Zenith blade which is a grab skill shot. It pulls Leona in the direction of the target her blade hits first. Minions are affected by this skill so try to go around them. Her R/Ultimate is Solar Flare which is an AOE ability. The ability hits anything within a radius from where you cast the ability. Any champion or minion in the epicenter of the blast will be stunned. If not in the epicenter, the target is slowed. Her ultimate must be timed correctly, if the champion is not in the epicenter, it would otherwise useless to cast the spell. Her Zenith blade has a long reach so use it wisely to initiate with her Q which follows after it.
You would like to chain her abilities this way: W, E, Q, AUTO, R.
R can also be used first or to finish a running away opponent.