I would like to start out by saying that AP Ashe the most viable option for Ashe but is tons of fun! I have had success with this build multiple times so I wanted to share to everyone. I do mainly play Ashe so people who are just starting out playing her probably should go a more traditional route before trying this build as it may be a little harder to get used too. People also may look at you funny for what your building so i like to let them know I'm going to build an Ap Ashe before the game even starts. (This is my first guide by the way, but I'll try my best)

This build may seem a little cray, but everything has its justifications. I would also like to point out that I tend to build my Ashe with a lot of attack speed to fully utilizes the frost shot ability,wit's end passive ( /black cleaver passive ) and the static shiv passive. Ill explain the items in the order you should build them. For boots I like to go boots of swiftness a lot of the times because Ashe is really slow and cc really hurts her so the resistance to slow really helps and it having more than average movement speed than other boots. Sorcerer's boots will work if you want magic penetration although this wont help you until you have built the nashor tooth item which applies bonus magic damage to your basic attacks. Wits end is a great item because it does an automatic bonus 42 magic damage with every auto attack and it also steals magic resistance of the target as you attack making your attack do even more damage. Next you go nashor tooth so you can start adding AP times. The nashor tooth takes 15% of your ability power +15 and adds that in bonus magic damage. So at this point your basic attack will do your regular ad damage plus 42 magic damage from wits end plus 15% of ability power+15 from the nashor. Building the static shiv next will add an extra 100 magic damage when the item procs and this damage can critical strike for even more magic damage and at this point you have tons of attack speed wish meaning it should proc alot. Now to start getting into damage items. Rabadons Deathcap add a nice 120 ability power and then takes your ability power and then takes 30% of your max ability power and adds it. For your last item Id go another Ap item. My personal choice is Deathfire Grasp which adds 120 ap and 10% cooldown reduction. Guinsoos rage blade has a good passive abilities. Black cleave does the same thing as wits end but with instead breaks down defense and not magic resistance(this would be good for support or just to do more damage with basic attacks). Will of ancients adds spell vamp. Blood thirster adds a protective shild and life steal. I dont really know why you would pick lich bane but you could try it.

Use frost shot to slow target and stopped them from running for the fight. Make your you toggle it off when not fighting champions.
Use Volley for cc and it also does a nice chunk of damage to the oppenents. Use to stay back from targets. Can be use when running from a fight to slow a lot of target and keep them out of reach.
Use frost shot to see into the jungle, so you don't get ganked. can also be used to see objectives like baron or dragon. It also has a nice passive to get extra gold form last hitting minions and taking towers.
Use Crystal Arrow to stop targets from running you. This slow also spreads to near by enemies. You can you it to initiate you may have to use it to kill. This ult shoots all the way across the map, but can only hit champions. You can get lots of crazy snipes with the ult to but this isn't recommended if you have no aim.

I feel this build goes both ways as does Ashe. Sometimes I support, but I mostly like to carry. A fed Ashe is the scariest thing alive. As a support you may want to build wits end and black cleaver to make it easier for your teammates to kill targets. As an Apc make sure to use your ult. Should do really massive damage because it scales on AP and not AD.

I go Ap just for the fun of it being different, but I also go Ap if i see were lacking Ap Champions. Its a nice twist which could be very good if played correctly.