Hello there,
I'm a Diamond 5 main jungler who has been playing a lot of rengar lately, and I figured I might as well try my luck with spreading the knowledge I've gathered over the months :) If you are already an experienced rengar player, you probably won't find anything new here, but if you are looking for an easy to read rengar guide for someone who would like to main rengar, you have come to the right place. Hope you learn something new.

Pros:
High mobility
Very strong duelist
Extremely good ganks post 6 and decent ganks without ulti with empowered Bola strike
Can 100-0 all squishies and burst down even tanks if ahead.
Requires map awareness and knowledge on target focus to master the playstyle.

Cons:
Can be shut down easily if focused/cc'd.
As pink wards are cheaper now, it is more likely that you get spotted while ulted.
No escapes other than ult and if taking damage ult will only activate after 5 seconds, which will most likely end up with you beind dead by then.
Requires an engage on the team to show his true potential as engaging himself will most likely be a 1 for 1 trade.

As jungle rengar, you want to rush warrior into mobility boots. Warrior gives you cdr and the early burst which you need. Yommus ghostblade is now an essential rush item on jungle rengar as the active and the cdr are amazing for ganking and general effectiveness. I have been getting hydra into death's dance or vice versa for my third and fourth item. I find getting Deaths dance to increase your survibability in fights and risky plays, although it is not the most cost effective item. A safe idea would be to get the hydra as third item as the active plays a key part in your oneshot combo. As for the last items, you can choose between the following:

Infinity Edge: This is my go to "NEED BURST" item as it increases your crit damage and sometimes one crit can oneshot an enemy. Since yomuus doesn't give critical strike chance anymore, this isn't as effective as it used to be. It is still a viable item if you are ahead.

Mercurial Scimitar: If you find yourself getting shut down by a heavy cc comp or simply them focusing you, this item is your best friend. It now also gives lifesteal which also goes well as a subsitute to Death's dance.

Deadman's Plate: This item works very well with Rengar's kit as it gives you bonus movement speed and more burst on your first auto attack. I tend to get this if I am against a heavy AD team and need the armour/HP to be able to stay alive long enough to deliver my damage output effectively.

Duskblade of Draktharr: If you are ahead early game, feel free to get this item after Yommus, although I find it unnecessary. The standard build has enough damage to oneshot your target. However, if you find yourself in a game where people are just barely getting away due to their mobility, this will fit in perfectly as a finisher. However, I recommend getting it only when you are ahead. Never get this when you are behind.

(All the below combos are when you start with 5 stacks of ferosity, which you should have before every gank)

For your pre 6 ganks you almost always want to enagage with your empowered bola strike as this is a hard cc ability and will allow your laner to follow up with ease, resulting in a flash or a kill.

Post 6, early game ganks you want to ult and use your empowered E during your jump animation, followed by a W, Tiamat while landing. Q upon landing. I like to save my second normal bola stike if they flash or use their mobility spells.

Post 6, mid to late game you activate yommuus just before reaching the target, activate your empowered Q while in range of the target and jump. You use your W and tiamat while in the jump animation and finish up with a normal E and Q.

You choose between the second and third combos depending on the target. If facing a leblanc, vayne, fizz or any instant dodge/mobility abilities I recommend using the first combo. Whereas if your target is a mage or someone you know whose spells/flash is down I recommend the second combo as it has the higher burst and you can stick to the targets if their spells are down. After a bit of practice this decision making will be second nature to you.

To use this strategy, get a smiteless leash at Gromp/Golems. Make sure you have 4 stacks or atleast 3 on your fury bar. Take E level 2 and make your way to the enemy buff as soon as possible. Make sure to let your team know earlier that you are invading, and to follow their laners if they react. Most junglers will be low and without smite when you reach them at their buff. Use an ability to get 5 stacks and empowered Bola strike to most likely secure the kill or blow their flash. As you have smite, you will get their buff 95% of the time. So this invade is very useful. However, I would not recommend vs healthy/fast junglers such as Diana as they would have cleared the buff before you get there. I also recommend to not take your E immediately, but to leave it open. If the enemy started top side for example, you can take W level 2 instead and simply take the buff as W level 2 is best for clearing.

6.17: Now the jungle clear is a bit different to what I used to. I start bot side always, and go golems>wraiths>wolves>gromp. If top lane is over extended then i gank, or else I back and get a long sword and a pink ward. Return bot side and do a full clear with buffs. This will give you level six and put you almost at par with xp with the solo laners. I try not to make ganks happen with rengar early, unless they are really extended or if you can easily counter gank with vision.)
I always start botside as you get a better leash. Smite the first camp and then take the buff second. You will be low after these two camps so taking scuttle and using your empowered W to heal is what I use to be able to complete the first clear. After taking scuttle, return to your jungle and take wolves and then the second buff. You can heal on the other scuttle and attempt to gank with 5 stacks as backing without 700 gold for the chilling smite if not very effective. Remember to keep 5 stacks on you whenever possible as it is a very important part of Rengar's kit.

When starting bot side, take the first camp smiteless and don't take an ability at level 2. Make your way as fast as you can to the enemy red/blue. Take E level 2 if the enemy is at the buff, wait for the buff to be low and attempt to kill/blow flash. If the enemy started topside, take W level 2 and just take the camp. This invade startegy should work in lower elos as very few people word their buffs and it is a lot safer as laners will rarely respond to help the enemy jungler. This works well against champions with a weak/slow first clear.

Works good against: Lee, Master Yi, Mundo, Wukong, Evelynn, Amumu and Xin zhao

Not recommended vs: Vi, Kindred, Warwick, Volibear, Udyr and Shaco.

Rengar has interesting synergys with other champions such as Orianna and Shen. You can deliver orianna ball for a sure multiple man ulti and shen ult can keep you alive as well as move him to taunt distance. If excecuted correctly rengar ult + Ori ult/shen ult can easily win a teamfight.

Having a hard engage such as malphite or leona also helps rengar as the enemy team would use their disengage on them which leaves you a window of oppurtunity to one shot who you please :)

Well, I hope you guys learnt stuff from this guide. This is my first guide, so have mercy :3
I don't claim to be a rengar pro and i'm sure there are better rengar players out there, but I have been having great success with this over the past weeks and just wanted to share what I had learnt.

Best of luck in the fields of justice, fellow summoners.