Why Should I play Riven?

Riven is an agile, fast hitting Champion, who is great at chasing enemies down, as well as escaping from said enemies. Able to slip in and out of combat, she makes a great assassin. Built properly, she can dominate the field in almost any situation.

Pros and Cons
Pros:

Very agile in combat
Good Damage
Very fun to play
Great CC for a dps
Great mobility

Cons:

Somewhat squishy
Doesn't deal huge damage til later in the game
Her ultimate is a skillshot, so you can miss with it
Gets focused a lot, as she does lots of damage

For items, you want 3 things. Damage, CD reduction, and attack speed. This is mainly because riven is an AD champion with no mana. The more CD reduction you get, the more attacks you can pump out. The more damage you have, the more those abilities do, and the attack speed pairs very well with her passive, allowing her to do the hits in between each ability for damage maximization.

Your Core items are definately your Tiamat, Wriggles Lanter, Your ghostblade, and your Trinity Force.
Tiamat early game is very useful for early game farming as well as it wreaks havoc in team fights. Since almost every ability Riven has is an aoe, this just makes you do more damage with your regular attacks. basically, when she's in a teamfight, she's always hitting everyone. With the addition with a modest amout of bonus damage and health regen, this is one very important piece of gear.
Wriggle is just awesome all around. Some armor, damage, lifesteal plus free wards make this a very nice item. I find that without the lifesteal, she's pretty squishy. So I take this very early game for the benefits and if I have money for it end game, I replace it with a Bloodthirster.
The ghostblade gives you a good amount of CD reduction plus damage and attack speed. Just what a good Riven needs.
Sword of the Divine gives great attack speed and the bonus 100 damage is not to be forgotten. all around a good item to have, just makes you more efficient late game.
Lastly you have your Trinity Force. You mainly get this for the Sheen around mid game, since it pairs AWESOMELY with your passive, doing crazy good damage. Once you get all your other items, you might as well upgrade it to a trinity, to get some more speed and some sweet sweet slow damage.
For your boots, you want boots of lucidity. Helps cap your cooldown reduction, plus some speed for chasing. Other good choices would be Boots of Swiftness or Berserkers Greaves.

Other items you could get would be things like malady for some magic resist instead of the sword of divine... Madreds Razor would be a good alternative to the Sword of the divine as well. Sword of the divine is really just a place holder for an item that you might need end game. If you need more armor and health, Sunfire Cape and Frozen mallet are good choices.

For Marks, I go flat Armor Penetration, to hit harder every time.

For Seals I go Flat Armor for more duribility.

For Glyphs I go for CD Reduction. By the time you finish your item build, you should have max cd reduction, making all your abilities available much more often, with your ulti down to an amazing 27 seconds.

For Quintessences put down more Armor Penetration.

Passive: Runic Blade. Every ability she does charges her blade. Basically a mini Sheen built in. To best utilize this, try to space out your abilities, allowing an auto attack in between each hit.
Your bread and butter ability. Does great damage, and has a knockback at the end of it. Very nice. It also gives you a small boost forward each time, so you can use it for running, or chasing. It gives you great mobility on the field as well, as you can easily switch between targets by using it to leap you closer to whatever your hitting. Since they do aoe damage, they are very good for taking care of minions as well.
Aoe stun. Does very good damage and stuns them. A good combo if chasing is to use up all of your broken wings to catch up to a target, and slam them with a ki burst. Often is just what is needed to get the kill on a target.
Great for chasing or running. The shield it gives is pretty much useless, but a shield is a shield. Can take about one turret shot, so you should use it. And since it procs your Passive, use it even if you don't need it, for the bonus damage your passive grants.
20% bonus damage for an impressive 15 seconds, plus a ranged ulti comparable to Garens, doing a skillshot that does more damage the lower the targets health is. Best strategy is to pop it once when you engage the fight, and use it again to finish them once they are low enough. Also, using the second portion of the ultimate does NOT get rid of the rest of your 15 seconds of bonus damage, so use it whenever you can do the most damage. With its low cooldown, be sure to use it as often as you like!

While there are several ways people say you should build riven skill wise, I personally believe that the BEST way to build her is by first getting your Q at level 1, your W at level 2, Q at 3, E at 4 and then level up Q and W alternatively keeping Q one level above your W. Get your R any time its up. Some people say that you should level your W and E first because they can get lower cooldowns faster. I however feel that Leveling your Q and E first should be priority, since they grant the most of your damage output, which riven needs early game. Alternating between your Q and W will keep both's damage relatively high, as well as lowering your stuns Cooldown very nicely. Your E is handy, but I find that I use it way less then Q and W, and it's not that important to me. It only is really useful once you have a lot of Attack Damage to boost its shield, and that doesn't happen till relatively late game,therefor I level it last.

My main choices are Flash and Ignite. Flash for obvious mobility reasons,such as ganking and escaping. Ignite is great for taking down people low on health, especially Tryndameres who have just used their ultimate. I usually can attribute 2-3 kills a match to a well placed ignite. Other Choices could be exhaust, ghost, teleport or cleanse.