Sivir is a champ I've recently tried out (for the past 30 or so matches?) and experimented a lot with. She has always been one of my favorites, but even I viewed her as kinda crappy...my opinion has changed somewhat.
She will have mana issues. Her range isn't good. She has no crowd control.
Big weaknesses that have to be addressed in order for her to compete.
Don't look at guides, because they almost all advocate starting with chalice or, god forbid, manamune, which costs a fair amount of gold in return for almost no damage initially (I understand manamune eventually does good damage, but are we really building for when we hit level 18 to be an impact?).
I'll just make this short, as I could go through my entire build and tell you why but it would take too long (also saw another sivir player with my item build but obviously not my runes, but her build was still good/decent).
I address her mana issues with per level mana regen blues and yellows. She has little range and thus needs some damage outright (in my opinion) early game in order to dominate her lane, I run attack damage reds and quint's.
For masteries I go enough in the defensive tree to get max level of vetern scars, and go enough in utility to get max level of mana regen and buffs.
I take flash/teleport as my summoners and start with a dorans blade. This puts me at a little over 600 life and a little over 70 damage starting. Those stats are good enough to handle ALMOST anyone mid. I have had problems against vayne as she doesn't really spam anything you can block, and her silver bullets will still out damage you. Against vayne, I would be tempted to start with cloth+5rp.
My core item build would be rush blood thirster after beserker boots and another doran blade if need be. Then get sword of divine. So when late game starts, your items will probably be...Doran Blade x2, beserker boots, BT, SoD. Your damage will be good. Now that late game is here, I try to get frozen mallet for the slow and life. Sivir needs a slow to keep targets from just running away, or to be able to run away. After mallet, game should be over, but if it isn't, just throw an atmas in there and you will be good to go.
DO NOT FORGET TO CUT SWORD OF DIVINE ON WHEN YOU FIGHT OR YOU ARE WASTING A GREAT ITEM.