EDIT: Added a DPS build for the haters and revamped for new chapter functions.

Something new that I've been playing around with. The funny thing is, it actually works!
The items work together to make you a powerhouse. You will enjoy a HUGE Health and Mana pool. In a 30 minute game, it's not uncommon to have 3000 Health and 3000 Mana. By the way, YOU ARE NOT A TANK!! Don't try to take on the whole team by yourself. You are still a squishy, just one who can take hits while carrying your team. You've been warned.

Pros

Great Soloer
Amazing Farmer
Only costs 450 Influence Points
Movement and Attack Speed Increase
Huge Damage Dealer
Deep Health and Mana Pool (with items)

Cons

She is very squishy (without items)
Mana can be a problem early (or you spam Boomerang Blade)
Usually a main target to focus (very focused when fed)

Abilities

Q: Boomerang Blade- Tosses blade for 280 (+75% of Attack Damage) Magic Damage. This is your nuke. Ungodly damage. Loads of mana to cast, so don't spam.
W: Ricohet- Regular attacks bounce to 5 additional targets, dealing 22% less damage per bounce (only first hit procs item effects). This is your farming skill. Never turn it off unless enemy is standing next to tower.
E: Spell Shield: Blocks an enemy ability used on you. Lasts 3 sec. When you block a spell, you gain 125 mana. Great for those stunners and gaining mana.
R: On the Hunt: Grants you and allies 25% Move Speed increase and 65% Attack Speed (half for allies) for 15 sec. Your teammates will love you for it. Also useful for getting the hell out of Dodge when things go bad.

As always, Ulty comes first. I like to keep BB and Rico at similar levels and max out SS last. Up to you to change for your play style though.If I'm going for the early gank (Rare), I grab BB first. On the Hunt (OtH) is taken whenever possible. I get Spell Shield (SS) second just to stop abilities for the extra Mana. Ricochet (R) comes third and is your main farming ability. After you get Tear of the Goddess, you should never turn it off unless enemy champ is standing next to turret or there is only 1 minion left to save mana. You should be able to stay in your lane forever by using your BB sparingly and good use of R. When the team fights start happening, your teammates will love your OtH. Extra attack speed and move speed is amazing. Always solo if you can. You can get gimped by laning wih a partner. With your items, your BB will hit like a freight train. No one wants to catch a BB n the face, especially if they are the only target it hits. Since, the damage is split when hitting multiple enemies, try to stand back and catch them at max range with BB. I like to stand way back out of range and run forward with BB to hit them and run back. It's annoying and demoralizing when they can't move forward without being hit. I like baiting champions like Pan by running ahead of the minions and into the range of their Stun ability. Pop SS and you have made them waste valuable Mana while you gain.

I use Flash and Teleport. Feel free to change to your playstyle. But please, for the love of all that's holy, don't pick something like Revive.

Runes are there simply for Mana Regen and some Magic Resistance. They are not set in stone and can be swapped to fit playstyle. I use Desolation for Marks and Quins for the much needed armor penetration.

Start out with a Sapphire Crystal and 1 of each pot. As soon as you get 595 Gold, go back to finish up Tear of the Goddess. Teleport back to lane and stay until you get Boots. Then rush The Zeke's. This gives you some damage while you work on your other items. I used to rush Warmog's after Tear, but THIS IS A MISTAKE!! You will not have the damage to compete with other Champions. These items are very potent when mated together. Warmog's give you Health and HP Regen. Atma's Impaler is there to take advantage of that Health by doing more damage. Plus, the armor and crit chance will help a lot. She be very squishy. Manamune lets you toss more Boomerang Blades thanks to increased Mana and Mana Regen and its also adds 2% of max mana to your attack damage. Zeke's for some attack speed, plus the life steal doesn't hurt. Phantom Dancer gives you attack speed, crit chance, and move speed.

Sivir is the queen of farming, so it won't take very long to max that Warmog's out.
I've almost ran out of things to say. You should have no trouble with any team except heavy CCers, and might want to swap Berserker's Greaves. The only static items are Manamune, Warmog's, and Atma's. It's a very solid build and I rarely lose if playing with decent players. I've even won a couple of 2v3 by waiting for them to enter base, killing them and rushing towers until they respawn. You will be a beast and they will fear you. Most of the time, the game is won by surrender because they are tired of feeding. Few games last all the way through. Well, I hope this helps you to be a better player. Good Luck and Have Fun!!! BTW, please vote and leave comments!!