Not Updated For Current Season
Sivir is a Carry / Farmer / Support type of character. She has a spell shield which would be good for blocking CC attacks such as Ashe's Volley.Her boomerang blade can destroy the whole entire minion wave if you aim it right. Her ricochet can farm really well early - mid game. You can also get a weakened enemy when they are trying to run away by hitting a minion close to the enemy.
Early Game: You want to get mid, get a meki pendant and HP potions. Sivir is extremely squishy early game. So pay attention to MIA so you don't get ganked. Try to tower hug as much as possible untill level 9, then you can start harrassing hard. You can bring atleast 1-3 of Ashe's hp down with 1 boomerang blade. Try to grab the Golem buff, Lizard buff, and kill dragon with your team. ( Don't forget to ward places :D )
Mid Game: By now you should have 3 items, Chalice of harmony, Bezerker greaves ( you can get Merc treads if there is heavy cc or alot of caster ) and a Starks Fervor. You want to start moving in with your team. When you guys get into a team fight hit your ult and try to stay behind the tank and just constantly hit them with Boomerang blade. Your ult can be good for chasing because of the movement bonus and the atk speed bonus.
Late Game: You should be getting a rage blade if not completed with your rage blade and getting a Hextech Gunblade. This is really good because the atk dmg AP ( which will help your boomerang blade ). You can now move around the jungle just using boomerange blade to clear up most camps. You can also gather your team to kill Baron Nashor. Baron Nashor will drop quickly because of the huge damage you deal with Boomerang Blade.
If you haven't gotten your Hextech Gunblade by 25 min you should go to top or bottom and just farm minions. Now you will be dealing a lot of damage. If you see your team mates getting chased get close to him / her and use your ult. The movement speed should help them escape. If they turn around then leave him / her cause then he / she is just an idiot.