Let me preface this guide by saying 90% of the information in this guide is from the Ultimate Riven Resource guide on solomid found here: http://www.solomid.net/guides.php?g=16914 It is the best Riven guide that I have ever seen and EVERY Riven player should be using it unless you are a pro. That said, since the author of said guide unfortunately has stopped updating the guide, I thought I would make this to keep it current with season 3 starting soon, and to provide some of my expieriences with the champion. I'm certainly not all that great at the game, but I like to think having played hundreds of games with Riven in every lane since the day she came out, I know enough to add some useful information to the guide.

Pros:

Incredible Burst
Dominates at level 1
Can go everywhere but bot viably, and can make a good kill lane bot if you're not being super-competitive
Extremely Mobile
Very few counters
Fun as hell to play

Cons:

Purely physical damage
No range
Using her spells for chasing severely limits damage output
Not much going for her as a jungler
Tends to be a bit squishy without her E

(Passive):
Most people tend to underestimate this passive, but it really is incredible. The extra damage on autoattacks turns you from just another AD burst mage to one with enough sustained damage to keep up your extreme damage until your skills come back up and once those are used anything worth attacking should be dead.

(Q): This is Riven's signiture move. It allows for great burst, 3 passive stacks, and great mobility all in one great skill. That said, I usually max it last since one point gives you access to the mobility and stacks, the CD doesn't decreas when leveled, W gives more damage, and E gives more bulk and more reliable jukes. However, if you really want the extra damage on Q you can still max it if you want.

(W):
This is where most of your early damage comes from. I max it first to lower the cooldown and up the damage, and once you get 2 or 3 points in it it becomes a great harassment tool. The stun portion in a 1v1 is pretty useless since it barely is any longer than the animation unless you cancel out of it with an E, but when more than just you is there it is a great AOE stun that is just long enough to help your team catch up to a fleeing opponent that only you can quite catch up to.

(E)
Without this, Riven would be very weak. The defence this skill provides is super important in 3 ways: 1. it increases you bulk WHILE you increase your damage, giving you good survivabiliy without having to sacrifice your high burst. 2. it gives you a quick, reliable juke to help dodge skillshots. 3. Despite not being able to go over walls, it gives a nice bit of mobility and even better, an unpredictable and fast enagement tool from mid range that isn't needed for your damage. I max it second for the CD, but against heavy harassment or range it is often better to max it first or split max it with W.

(Ult)
Why Riven is Broken in a nutshell. This alone turns you from a threat to an all-out MONSTER. The massive AD spike gives you a massive damage increase that scales into the lategame AND increased bulk from your E, and the execute allows you to burst people down in mere seconds. Best of all, the low CD allows you to pop it whenever there's a major fight at rank 1, a teamfight at rank 2, or basically any time you attack anything more dangerous than a buff camp at rank 3. Make sure to use it as soon as you can during a fight since it boots all your other abilities, but don't use it too early during ranks 1/2 and waste it when the enemy backs off.

I run these runes primarily to increase my earlygame but I also have the MR/lvl blues to help survive enemy burst lategame. If you aren't very worried about the enemy APs or they are running a primarily AD comp, you can swap them out for more damage if you perfer. With these runes you should be able to A. Duel well with or burst down most AD tops earlygame (I say most *cough* Garen is a pain *Cough*) B. Burst down most squishy AP/AD carries in bot or mid. You might be thinking "Isn't Armor Pen better", but usually early on the pure damage runes give more results and lategame you wreck people regardless unless they stack armor, in which case just build a Last Whisper and/or Youmuu's

Pretty standard 21/9/0 except that you need the CDR since you are so ability focused. I get the four by not taking Lethality because you never build crit and Vamparism since a Doran's Blade is far more helpful for lifesteal before you get your Vamp Scepter. The nine in Defence is pretty much personal Preference so put it wherever you want. Personally I like having the early armor for top or MR for mid and the extra health for early dominance and the Tough Skin for less damage from minions when you fight or harass behind their line.

/ is pretty standard. Flash is the best spell in the game 90% of the time and should always be used. You can run Ghost if you really want but flash is almost always better since you don't have trouble sticking to people. Ignite is very helpful for getting kills at all stages of the game, but if you want to run Teleport for ganks, or Exaust/ Heal to quickly turn a fight you can. Obviously, run instead of ignite when you jungle so you don't lose every buff, dragon, and baron in the whole game. Anything beyond those is pretty much unviable.

First point is almost always Broken Wings since it provides incredible mobility and damage. Sometimes against heavy harass ranged attackers I start Valor but that's fairly rare. From there usually a point in Ki Burst is the way to go, but if the enemy's harass is too much to allow you to get close or farm well, get a point in Valor and max it until the problem is remedied. Otherwise, max as written above and it will be pretty obvious if you need an early point in Valor.

Riven is usually a great pick for any role but you need to make sure you don't leave any holes in your comp.

Obviously, don't steal the role someone else called or picked before you. (I and most people I meet follow the Call>Pick rule, don't waste your breath arguing that it's the other way around.)
You need a tank, and Riven is NOT one of them. If you go top/jungle make sure the person who is doing the other role is building tanky and playing a champion who works as a tank. If you go mid usually someone else will and it will rarely fall to you.
You need AP. If you go mid, make sure either the jungle or top plays an AP champion. You don't want the enemy stacking armor and never dieing. If you are top, make sure your mid isn't Zed or Talon.

You can go pretty much any lane except bot in a competitive game so as long as your team is solid go where you want. Top I find is the most useful to the team and is where I go most of the time. Jungle Riven works fine, but I feel that she doesn't do anything much better than other junglers since her clears aren't bad but aren't nearly as fast as udyr or shyvana, her ganks are only good when the enemy is pushed or you have high CC in lane, and she isn't great at counterjungling. That said, her lategame is still as great as ever so if top is called go ahead and jungle. Mid is a bit different from Top and Jungle. First, the lane tends to be VERY volitile, if you don't burst down and destroy the enemy laner, they usually do that to you. Secondly, your role is not so much to wreck your lane (Although if you can definiely do it), but moreso to wreck the other lanes. Your ganks may not be that great, but your high burst and mobility allow you to help your jungler snowball the other lanes while also getting fed yourself. I wouldn't reccomend Midven to newer Riven players, and most lower elo players will dodge or report you when you lock in, but it certainly gets results when it works.

Recommended Build
Starting out you can grab either /5, /3, or . It mostly depends on who you're fighting. Against an AD with heavy harass or a good early duelist like Garen or Fiora, start cloth. Against someone you can easily burst down level 1 or 2 like Vlad or Nasus, start Doran's. Against everyone else, start Boots. When jungling start boots since you usually don't need cloth/5 for a full clear and you often don't build it into anything. The one exception to the above guidelines is if you are 100% sure there will be a level 1 5v5 teamfight (i.e. you are invading). In this case, always start Doran's since level 1 Riven with Q and Dorans has the most damage output of any champion in the game at level 1 with the exception of Rengar. The only time this is not the case is if you are fighting someone like Garen who can CC you. If you don't start boots get them as soon as you can unless you are so fed after your first B you can already buy a BF sword. After that rush the BT since it is your core item and where 99% of your damage comes from, usually staying in long enough to get your BF sword and saving for it if you B beforehand. If you really need the early game dominance you can also get a doran's or 2 first if you need them. I don't often get the Vamp first because it delays the BF but if you are getting harassed a lot get it to help stay in lane. Why no Brutalizer? Well, the thing with brut is that it only helps you win early to mid game, and 9 times out of 10 you either are already winning your lane and it would be better to just get the BF sword, or are losing too badly for brut to matter much. It also doesn't build into anything useful since Youmuu's has so much gold wasted on Crit and the active which you don't really need. If you really want it or the lane is very even, go for it, but I find it's only about 300 away from BF, I just save up. The one thing to note about it is if that Van Damm's Piliger from TT makes it's way over to SR then it will become Riven's core item since EVERY stat on it is extremely helpful on Riven. The one major item I didn't mention above is Heart of Gold. If you are losing your lane or have no hope of killing your opponent because they are too tanky, always get the HoG ASAP. The extra gold really goes a long way to outfarming your opponent. When jungling it's usually a good idea to get it early as well since your income is usually lower than when laning. After you get your BT unless you are getting fed you should usually grab a HoG since your next item, GA, is so expencive. From here on out, upgrade your boots whenever you want, usually I do it before GA or right after, depending on who I'm fighting and how much gold I have when I B. Generally always get a core item if you have enough for it when you B, if not upgrade boots. If you are against an AP or the APC is fed, get mercs, if you are against AD or an AD is fed, get Ninjas. When you are jungling if you find yourself ganking nonstop and not hurting much you can get Mobies. No matter what, switch Ninjas or Mobies in for Mercs lategame unless the enemy has extremely small amounts of CC or the enemy ADC is the only one with very many kills. Anyway, after BT/HoG/lvl2 boots get the GA ASAP since you often have a high tendency to die in teamfights and your high mobility and instant AOE stun can sometimes let you get out of sticky situations after you res. After you get the GA, ask yourself one question: Am I fed? If the answer is yes, ask: Has my GA been popped more than once for every 20 mins of the game? and If I leap into a teamfight, if the enemy team focusses me am I likely to die? If the answer to all of them is yes, stack BTs until you win or an answer becomes no. If any of the answers is no, read on. After BT, GA, and HoG, your build becomes situational, if you are mid, against and AP top, or the enemy AP carry is fed, grab a Hexdrinker and upgrade it into a Maw if the APC is still scary. If the enemy AD champions are fed or noone is fed, get a Warden's Mail and upgrade it and HoG into a Randuin's. After one, get the other, and after that you can get whatever, usually I get another BT since I'm usually not dieing much at this point. Usually most games will be over by the time you get your second bulk item.

Other Options
Youmuu's: As I said before it isn't that great, but if you get a brutalize you may as well upgrade it after your first bulk item.
Wriggle's: When you jungle you might want this to help clear better and the free ward is always nice. However, like I explain later on in the Jungling section, you don't really need it if you get a lot of good ganks, and if not double doran's is pretty much the same bang for less buck. Plus I'm an idiot and never remember to ward anyway so the free ward doesn't help me much lol.
Warmog's: Usually I like Randuin's more since it gives armor AND health, upgrades from HoG, and the active helps stick to people when your stuff is on CD, and Maw since it increases your damage by a lot, but Warmogs isn't a bad option to consider if the enemy is evenly fed across AD and AP.
Last Whisper: Get one if the enemy has: Taric/Sona, Rammus/Malphite, more than one player with more than one armor item, or a tank who really hates you and is stacking armor.If not, another BT is better.

What NOT to get
Phage: It may be reccomended by Riot, but it really is TERRIBLE on Riven. In terms of stats, it is inferior to the above in both damage AND bulk. Also, the passive may seem nice, but since you have so much mobility and burst anything that can survive all your abilities and a few autoattacks and can still get away should not be focused anyway. Plus Randuins gives you a quick slow anyway.
Frozen Mallet: Same as Phage
Trinity Force: Oh wow, a sheen proc. Soooooooo helpful. Actually, with your passive a sheen proc would be nice, but it wastes incredible amounts of gold on useless stats. Let's break it down shall we? Health: Helpful. AD: Helpful, but inferior to BT and Maw. AP: No use. Mana: No use. Crit: A nice gesture, but not that great since you don't autoattack much besides when u have passive stacks. Attack Speed: Same as Crit. As you can see, it provides only 2 useful stats in mediocre amounts, 2 useless stats, 2 barely helpful stats, and a semi-helpful proc. For the highest gold amount for any item in the game? Sure as hell not.
Infinity Edge: It provedes less damage than a fully stacked BT, crit isn't very useful, and costs more money than BT. Pass.

If you want to see anything else, leave a comment and I might put it in somewhere.

This is a work in progress, so it will be added to as I go. If you don't see a champion you have fought more than once or twice here, leave your elo range (general range, I really don't care if you're 1k or 2k, just low, mid, high), how you did, and what you built or thought you should build.
Been typing for an hour, can't be bothered to put anything in yet, check back later.

Path
Take wolves before blue spawns, then take it. Help with damage on wolves and leash on blue recommended. Don't be that guy and tell your mid to leash wolves and lose half his health. Then go wraiths/golems/red. You can gank anywhere in here if you want, just make sure you have W when you do.

Build
Mostly the same as above, but you might want early damage to help clear. You could get Wriggles like most people, but I find being a hipster and getting Double Doran's is better. Overall they give you the same damage and bulk, but while Wriggles gives you more lifesteal, it also costs a lot more, so going doran's and selling then later helps you get your core faster. Always start Q, but from there your next 2 points can be put wherever. A point in E at level 2 alows you to clear with more health if a bit slower, and a second in W at level 3 allows you to clear red faster. One thing that separated the Provens from the Noobvens is using your Q at an exact moment so you use the last hit on a camp and the cooldown is up a lot faster (since the CD starts at the 1st activation, not the last). I could never explain it better than this guy, so look at the Jungle Paths section of this guide: http://www.solomid.net/guides.php?g=16914

Ganking
Riven's ganks aren't that great so you need some help:
Get your lanes to tell you who has warded and where, and make sure to get oracles/pinks and clear them out when you can't gank but are in the are. Also, if the lane has CC, have them lead in and CC them. If you can get to the enemy without needing to use your Q, they are probably dead. Fight how you would normally, but make sure that the last hit of your Q goes BEHIND the enemy since it has a slight knockback. If you manage that then they are almost guarenteed to die or pop a summoner.

Riven is an amazing Champion, and I love every bit of her. I hope this guide helped you get a bit better with her, or at least point you in the direction of how to get better. Did I miss something? Did I think something was obvious but isn't? Leave a comment, I'm always open to feedback! (Constructive critisism plz, suggestions in the form of "YOUR ****IN STUPID YOU SHOULD HAVE SAID THIS YOUR A ****IN ****** UNINSTALL" will probably be ignored.)