Probably the only controversial choice here is HP Quints > ArP Quints. IMO, HP quints let you lane longer and farm more, which I think is more useful. With ArP Quints, you're just a little too squishy early game.
To play Ashe well you, have to get good at laning and last hitting, while harassing your opponent. At the beginning of the game I like to buy my items (Doran's Blade, Health Pot), quickly and run straight for the mid bushes. This lets you immediately take squishy champions to half health with a Volley + Critas soon as they show their faces. If you take them low enough you might be able to start zoning them early, which basically just wins you the lane right there. You want to play conservatively and not let yourself get harassed by enemy champions. This means staying at the edge of the creep fight and last hitting, and only getting close in order to throw a volley at them. As you approach lvl 6, try to have enough mana for an Arrow + Volley. This will almost always either let you get the kill, or send them back to base, giving you about a level on your opponent. Even better is if they stick around, which means you WILL kill them as soon as your ult refreshes. My item build is very expensive so make sure to always make as much gold as you can. For instance, if you ever find yourself not knowing what to do, just farm some neutral camps.
On my first trip back I aim to have around 1500 gold, enough to finish your boots, and buy a second Doran's Blade. If you find yourself with 1850 on your way back though, you can definitely skip the second Blade and go straight for a B.F. Sword. On my second trip back I try to have enough to finish I.E., but failing that, try and pick up one or both of the other two pieces. After that just go back when you need to, but try to finish part of an item every time you B. Banshee's will save your *** countless times with Ashe, as one of her biggest weaknesses is getting stunned and killed. Some people may be confused at my last two items, but here's my reasoning. The rest of your items are quite expensive and if you have completed all of them and the game is still not over, it probably means there's some huge roadblock in your way, (i.e. a fed Rammus). Black Cleaver lets you help your team take care of huge tanky champions. By this point, the game is usually over. Usually, the only time when I find myself holding a Phantom Dancer, is when the rest of my team sucks. If you're fed enough for this item and you're team still hasn't won the game, someones feeding your opponents a LOT. Phantom Dancer lets you dish out more pain, and let's you run around the map and kite like a pro.
One important thing about Ashe is to use Hawkshot effectively. Use it to peek at Baron, Camps and Dragon as well as the side bushes while you're laning. Try to use it as often as possible.
Try to always keep Frost Shot in mind. In general, after level 10 or so, just toggle it on to hit champions, toggle off for creeps. It's not that tough.
Try not to use your ult to KS. I know how tempting it is.
Try to bait with your ult. In lane, you can let yourself get a little low on health in order to get your opponent to pursue. Then you can just ult them in the face and destroy them. I've been in a few hilarious chases where I'm running around the map, dodging skill shots and ****, while being chased with like ten seconds left on my ult. Then just turn around and kill them. Another thing to note is if you die or B, I like to go through the jungle back to my lane, letting them push up to my tower. Then just lay down Ult, turn on Frost Shot and Volley/auto attack/crit them into submission.
One hilarious thing you can do is shoot squishy champs that are smacking your tower with just a few enemy creeps backing them up. If you shoot a long range Arrow they'll be stunned for 3.5 seconds and the splash will kill the creeps, so the Tower will aggro them and they'll die.
This last thing takes a lot of practice, and i'm still pretty bad at it. People will focus Ashe given the chance, but with the right team you can bait them into chasing you but just fire some frosty Volleys to keep them at bay while your team pursues. This is where Flash comes in handy, as you can often just run into a bush and then flash behind your team to safety. I usually only try this when I have a Banshee's up though, so a stun doesn't just kill you.