While some might have played manaless champions before, Renekton's Fury resource, (now shared with Tryndamere and Shyvana), can catch many players who have never fought or played as Renekton off-guard. While he needs fury to enhance his abilities, he does not need it to cast them. Try to use this to your advantage.
Although they aren't on-next-attack abilities, Cull the MeekCull the Meek and Slice and DiceSlice and Dice also reset Renekton's autoattack. So try and weave all his abilites between autoattacks to increase your DPS substantially. These autoattacks also generate fury, so do not neglect them!
When you need to make a quick escape use Slice and DiceSlice and Dice into a minion wave or champion. This will trigger the secondary effect, which allows you to use it again, clearing a good distance very quickly.
When escaping through the jungle, you can use Slice and DiceSlice and Dice to go through some walls. Hitting monsters on the other side of the wall will allow you to cast Dice letting you dash through another wall, making this maneuver very effective.
Slice and DiceSlice and Dice can as well be used to harass enemies who have their abilities on cooldown. Slice in, land a few hits and make sure you Dice back out of range in time before the enemy champion has time to react properly.
Frozen Mallet is a very good item for Renekton as its movement speed slow allows him to stay in range and constantly use his abilities. Plus, it gives him some much needed health, as he is pretty squishy early and mid game.
Dominus is an extremely powerful ability, especially in the early to mid stages of the game. A Renekton in this form is more durable than most champions, and can enable overwhelming amounts of damage, essentially being able to kill two champions by himself easily. Approach a level 6 Renekton with care.
Renekton might choose to build more towards damage or tanking. Pay attention to this: If you force him to build tanky he will greatly fall off late game because of the nature of his kit. If he builds towards damage, prioritize him accordingly in teamfights and be sure to lock him down with crowd control. He has extremely high ability ratios but low survivability outside of health sustainment and Dominus.
While tanky dps Renekton greatly loses power late game, he's very difficult to deal with early to mid game. You should try to isolate him out of teamfights and always know his location.
Renekton has an extremely strong laning phase and very high early game damage because of the high base damage his abilities have, especially their Fury Empowered versions.
Never get careless against him: He has a powerful combination of a stun, mobility and sustain.