AD = Attack Damage
AoE = Area of Effect
ArP = Armor Penetration
ASpeed = Attack Speed
CC = Crowd Control
CDr = Cooldown Reduction
DPS = Damage Per Second
HP = Hit/Health Points
KDr = Kill to Death Ratio

Welcome to my first build guide on MobaFire!
I first played Renekton during free week, and he quickly became my favourite champion, because of his amazing damage output and killing ability. The first attempts I had at building Renekton were based mainly around AD items. Whilst this was an awful lot of fun, it tended to do bad things to your KDr. It also made him exceptionally useless towards the end of the game as he would tend to die quickly, owing to the fact that he has to be in close to put out damage (and the sheer amount of damage output an AD based Renekton could put out meant you were usually focused first).
So I put more and more tank items into the build, and whilst his abilities still did great damage they were all too long cooldowns for his damage output to be sustained, and not quite enough damage for him to be considered a nuke. So after some deliberation, I came up with this build, which balances tank and damage effectively and has max CDr to allow his abilities to be spammed, which is the main source of his damage.

Pros:
Great harass early game.
Good tank/damage ratio.
Fantastic damage output in teamfights with Dominus and items such as Youmuu's Ghostblade.
Great farmer/pusher.
Cons:
Early-game lane dominance can make you feel a little more invincible than you are.
Easily shut down by Exhaust and other such effects.
Can't put out much damage if you're denied from building Fury.

Reds/Quints
Greater Mark of Desolation & Greater Quintessence of Desolation
Flat ArP. Makes him hit harder, which is never a bad thing.
Alternatives:
Greater Quintessence of Health
Flat health quints can improve his staying power early game, however I prefer to hit just a little harder.

Greater Seal of Evasion
Dodge chance is always nice to avoid those pesky auto-attack heroes, and the point in Nimbleness can help when chasing/escaping. One nice side-effect of this is when you are harassing in the lane early game and you draw minion aggression, chances are you'll dodge one of the attacks and you can chase/escape all the better.
Alternatives:
Greater Seal of Health
Flat health can also help to keep you in a lane longer early game, however I prefer a heightened chance of the Nimbleness buff.
Greater Seal of Armor
A viable alternative if you don't have the IP for a full set of dodge/health yellows.

Greater Glyph of Scaling Cooldown Reduction
CDr. CDr = more ability spam, and more ability spam = more damage output. Simple.

I play 8/21/1 Masteries, as described above. You can take the point out of Haste and place it in Sunder, however I prefer the extra speed and duration from the ghost buff for chasing down/escaping from opponents.
I have tried 21/9/0 masteries before, and while this does allow for a lot more damage, I find that you're a much easier target for harass early game and you don't really get to apply that extra damage.

I like to take Ghost for the utility to chase and escape. When combined with Slice and Dice you can move much faster than a lot of champions.
Exhaust again is there for the chasing/escaping abilities, and the reduced damage can give you an edge in engagements.
Alternatives:
Cleanse can be used as an alternative to ghost, however if you're still in range of enemies the chances are you'll just be hit with more CC effects. However if you can use Slice and Dice to put range between you and opponents, it can be useful.
What I don't use:
Flash - I've seen a couple of Renektons use flash to close a gap between them and a champion and allow them to put Ruthless Predator onto an enemy champion. However, I use Slice and Dice for this purpose and thus I find it quiet redundant.
Ignite - The extra damage is nice, however I don't think you should be relying on it to finish off champions, so I never take it. Can be useful against characters like Fiddlesticks and Swain for shutting down their regeneration, but personally I find this too situational.
Teleport - You can move pretty damned fast with Youmuu's Ghostblade and Ghost, so I find it redundant. Also, it reduces chasing ability by rather a lot.
Fortify - Whilst nice for defence, the huge cooldown and removal of a chasing ability means I never pick this.

I take a Cloth Armor and 5x Health Potion to start, as this gives great staying power early game, and can save you from heavy harass. When you have the gold, upgrade it to a Heart of Gold for a health boost and some extra Gold/10.
Second item should be Ionian Boots of Lucidity. The CDr that these boots provide is a huge early-game damage boost.
Next, The Brutalizer. It gives AD, ArP and CDr rolled into one, which is everything Renekton needs to keep putting out damage.
Then a Kindlegem which should be built into a Spirit Visage as soon as possible. This item gives CDr, MRes and HP, and a healing buff which increases the health returned from Cull the Meek. This is a great survivability item for Renekton and should be bought as soon as you have the gold.
Next, I change The Brutalizer into Youmuu's Ghostblade. This increases ArP, AD and CDr and also throws in some Critical Chance, all of which increase your overall DPS. The active effect is great on two fronts - the movement speed buff allows you to chase/escape more effectively, and the ASpeed buff allows you to build up rage quicker, which empower Renekton's abilities massively. I use the active effect of Ghostblade in almost every encounter I get into, due to the huge increase in damage output. A great item overall.
Once you have the Ghostblade you should set about building a Randuin's Omen. This gives HP, HP/5 and Armour, all of which increase survivability further. Also, the move/attack speed slowing effect are another great chasing/escape tool (especially when paired with Youmuu's Ghostblade) and can turn the tide in team fights if the enemy team has a lot of physical damage heroes.
Once you have bought Randuin's Omen, you have enough spare CDr to drop the Ionian Boots of Lucidity and pick up a pair of Mercury's Treads. The CC reduction and MRes can potentially add a fair bit of survivability, and reduce the effects of being shut down by Exhaust and other such effects. You can, if wanted, pick up a pair of Ninja Tabi instead, however this is only effective if the enemy team have a lot of AD heroes. 9 out of 10 times I find myself picking the Treads for the CC reduction.
Now, at this point, Renekton's damage can start to feel like it's arcing off, so I grab a B. F. Sword and build it as soon as possible into Black Cleaver. The Black Cleaver gives an ASpeed buff, which allows him to build up rage quicker, and the ArP debuff is great for decimating enemy champions. A great bonus of this item is that the ArP debuff is applied by Ruthless Predator, meaning that it will apply all 3 stacks instantly if Renekton is enraged (has 50+ fury).
If a game is lasting a long while (which it is if you're this far in the build) I grab a Force of Nature to further increase survivability. An advantage of this is that Renekton can regenerate his entire health pool incredibly quickly, allowing for sustained pushes (that will hopefully win you the game). The MRes also rounds out his resists nicely, meaning you are not particularly vulnerable to any champion's attacks. And lastly, the move speed buff is great for chasing down enemy heroes.
If you've run out of things to build, then just keep grabbing Elixir of Fortitude and Elixir of Agility with your spare gold, and Oracle's Elixir if they have a stealth champion, or a Teemo (because ******* Noxious Trap is irritating).

DUO LANE:
I take a point in Cull the Meek to start, as the lifesteal it provides is a great way to stay in the lane early game. I would not start trying to harass until I get a point in Slice and Dice. This is a great harassing tool, as you can slice through a minion wave to an enemy champion, hit Cull the Meek, and dice back out again to safety.
If you find yourself on less than 50% health when in the lane early game, auto-attack creeps until you have 50 fury (provided you can do so fairly safely, do bear in mind that you can use Slice and Dice to get out of range of harass), then slice into the middle of a creep wave, hit Cull the Meek, which gives a great amount of health, and dice back out again. It takes a little practice to do it in one fluid movement, but when you perfect it you can sustain yourself well in the lane early game.
When you get a point in Ruthless Predator the pain can really start. Build up 50 fury (or if possible, 100), slice through the wave and hit Ruthless Predator to stun an enemy champ, hit Cull the Meek and then slice back out. Provided you can pull this off fluidly it's a great way to wear down enemy champion health.
When you hit level 6 and get a point in Dominus you can generally go for a kill (provided the harass has been going well. Dominus turns Renekton into an almost unstoppable killing machine, as the fury per second gives you loads of empowered abilities to spam which makes for huge damage output. Also, the health bonus gives great staying power and often provides a nice edge over opponents. Throw in the AoE damage and it's an incredibly effective tool for killing opponents. I usually pop it every time an enemy commits to a fight as it will usually spell death for them.
SOLO LANE:
I use the same skill sequence in the solo lane, however I play much more conservatively, only moving in to last hit creeps (if safe to do so). Farm as best you can, only harassing one of the champions is off-lane, or there is enough distance between them so that you can be in and out before they can react. When you get a point in Dominus, you become incredibly potent, as the enemy champs should still be on level 4/5. Their lack of an Ultimate (which usually translates to damage output) means you can at this point go in for a kill. However, I would only advise trying to pull this off if you already have a feel for Renekton, and you're playing against two champions who don't have a huge amount of CC.
MID/LATE GAME:
Generally, with Renekton's ability to sustain himself in a lane and his great killing ability means I can usually have the tower pushed fairly quickly. At this point I leave the lane and go and help push/gank other lanes, whilst diving in and out of the jungles to farm and grab the Lizard buff.
Now you should be in the mid-game phase, and Renektons survivability matched with the damage output means you are truly a force to be reckoned with. Unfortunately this sometimes has the side effect of me overcommitting to stupid situations (for example: I have been known to dive under a tower that has 4 champions on it, simply because one of them is on low health and I decide to try my luck. 50% of the time, I die). At this point I generally move with the team, looking for easy kills and pushing opportunities.
At his point you are usually also tanky enough to initiate fights (if your tanks aren't initiating, that is. You are by no means a true initiator, it's just an option you have). However, because of the distance you can cover with Slice and Dice it's easy to leave your team too far behind and get destroyed before they catch up (another move I've been known to pull off).
When a teamfight gets properly initiated pop your Randuin's Omen, Youmuu's Ghostblade and Dominus, then try to focus the squishy characters with your basic attacks. Spam Cull the Meek and Ruthless Predator. You can spam Slice and Dice early in the fight, however I like to save it later on to chase down anyone who tries to leave the teamfight with low health. The AoE damage from Cull the Meek is great for wearing down the team, and when Ruthless Predator is available try and put it on a high-DPS character. Generally, the AoE damage from Cull the Meek and Dominus can really help to win teamfights. At this point there's nothing more to do that continue the flow of Teamfight > Push > Teamfight > Push, hit Dragon and Baron Nashor if they're up, and (if all goes well) destroy their nexus.

I don't make any promises that using this guide will make you immediately great with Renekton, as there is a learning curve associated with learning to get a good flow between your abilities, and learning when to wait and when to strike. However Renekton is a great champion who is hugely rewarding to learn how to play, and a hell of a lot of fun when done properly.