sivir is a very tricky build. you have to be super careful early game especially. this build is essentially the best way to survive on sivir minus stacking armor and warmogs...
but here we go
the first thing that you must realize is that sivir is a SUPPORT!!!
u should not run up into the middle of a team fight and immediately die. stay in the back and let the tanks do the work. just sit in the back and try not to get hit. then usually about 30 mins into a game.. that's when u can do work!so bide your time and be patient.

SKILL-
boomerang blade- a little known fact, sivir has the biggest nuke at lvl 9. your boomerang blade can half life just about any mage if u hit them at the very end of the range. try to get a clean shot at someone with the least amount of minions and other champs in way. be careful early game and don't spam to much. only use on necessity not to farm.

richochet- her trademark ability. use it to farm. start building it up but dont activate till you get the manamune or you will have 0 mana.

spell shield- this is my favorite ability. it negates one ability used against you. if it blocks an ability it gives u 150 mana back. which in turn, is another boomerang blade. this is why i dont run mana runes either, beecause if done correctly u never run out of mana. i try to get 2 points in this at lvl 2 and at 4 if im going solo mid as well. it gives u more mana and that means massive damage early game and great for ganks against the people 2-3 lvl lowers than you.

on the hunt- well it does 2 things. give u increased attack speed ( which after madreds when u pop this your att speed should be maxed) and movement speed. so in turn it give u the great ability to either run or stand and fight. i like to use it on turrets personally so i can bring them down insanely fast.

summoner abilitys
heal- early game use for sure. sivir is SUPER squishy. this even outs the playing field so you can push more and still survive.
teleport- sivir can defend a turret by herself. since you can tele to minions it can get you to a team fight or for a back door turrett kill.
exhaust- fun to use. even though you really shouldnt get close enough to anyone to use this. just try to keep them back.
ghost- this can save your on the hunt for just fighting and you can use ghost to flee or run to something
flash- ^^^^^^^^^^^^^^^^^^^ lets u escape lol

early game-
i always start off with a long sword. mid or not it is usually the easiest way to get minion kills and if your lucky u can even get a champ kill.try and lane with a tank. if you cant then thats okay because you are still one of the best early game harassers. plus if you have heal you can still run up and make a mistake and hopefully still survive. that being said, DO NOT SPAM BOOMERANG BLADE!! it is a waste of mana that you can use for either your spell shield or for last hitting a champ that is running away. just try to lay low and bide your time until you can get manamune. try to last hit minions and work your way up to 1k gold. once you get 1k gold immediately get tear of the goddess. dont be afraid to buy another hp pot if you need it. staying alive is the absolute best thing you can do on sivir. then wait 700 more gold and buy manamune. this is when u can finally start using ricochet. i usually have my manamune by lvl 9 with no champ kills and 0 deaths. so it doesnt take long. and now you are essentially a farming god. once you grab berserker greaves you can enter mid game.
mid game-
this is the easiest part of the game. FARM FOR GOLD! the faster you get starks and a blood thirster the better you can be. always get recurve bow first then upgrade to starks. this way you can farm faster and do a lot more damage in that team fight that is bound to happen. once you do finish off starks then you can solo drag and any of the monster buffs with ease. so the first thing you should do is get those. then save up golds up for a bloodthirster and once you complete that... u can 1v1 with most champs - a tank in the game.
late game-
this is the part where you start to get to have some fun! you should be doing close to 200 damage at about 1.3 attack speed before your ultimate. so when that team fight does pop up you can last hit people and harass with the boomerang blade and on the hunt. at this point if done correctly, you can hold off an entire team if you are on a turett. which is why i run teleport. so when you see a bunch of people pushing a turret tele up there and try to nab a quick kill. work your way up to madreds bloodrazor then it is essentially good luck to anybody. now after madreds is where i go team specific. if im getting hit hard and cant run away as fast get a frozen mallet. if its a lot more squishy people then i go another bloodthrister. if its a tank heavy team i try to get a black cleaver. after the madreds though you can do A LOT of damage super quick even before your ultimate.

conclusion-
sivir is so much fun to play. shes an undervalued support champ who can deal out insane damage if built right. so if u enjoy playing underpowered champs and wrecking op chos 1v1... this is the right champ for you! good luck and have fun!! =D