Strategy and Tactics

Strategy

Jungle Routes

Due to Rengar's strong level 2 ganks and not being reliant on Blue, he has various openings.

Path 1: Level 2 Gank
Route: Red (Smite) -> Wraiths -> Gank Top/Mid -> Wolves -> Blue(Smite) -> Shop
Skill Order: Q -> E -> W 

This path attempts to gank very quickly using your passive to close a huge gap followed up by high damage and long crowd control, as well as Red Buff.extremely strong level 2 kit and the strength of Red Buff. You are able to get kills really easily or force summoners and take them to really low HP as long as your ally follows up.

Path 2: Level 3 Gank
Route: Blue -> Wolves -> Red (Smite) -> Gank Top/Mid OR Wraiths -> Red -> Blue(Smite)
Skill Order: Q -> W -> E

This path is similar to the first path where you are aiming to gank very quickly however this gank is much stronger but comes a bit later. The most important thing to note is you need a really strong leash on the first buff you do in order to save Smite without losing much time or HP. Starting at Blue or Red depends on what side you are, and which lane you want to gank.

Path 3: Just Farm
Route: Blue -> Wolves -> Wraiths -> Red(Smite) -> Golems -> Wraiths

OR Red(Smite) -> Wraiths -> Golems -> Wraiths -> Wolves -> Blue(Smite)
Skill Order: Q -> W -> E -> Q

This last path just focuses on getting 4 while sustaining HP. You can start on either buff since you aren't dependant on blue buff. You are aiming to just farm till 4 then pressure the map with this route. You want to do this path when you do not expect to be able to kill certain lanes early as it will be pushed to their side or they are just difficult lanes to gank. With this skill order, you are able to choose to start ganking at level 3 with a point in every skill, without damaging your jungle speed/sustain by much.

Early Game

Due to Rengar being manaless, he is able to start at either Blue or Red side. He is also able to gank from level two if you get Bola Strike level 2. Thus you should always be looking for ganking opportunities, just make sure you are going in with full ferocity. Lanes that have opportunities to gank are lanes that are pushed towards your side, have summoner spells on cooldown, and your laners have sufficient resources (HP, mana, cooldowns) to follow up on your gank. Lanes that need ganks are lanes that are behind either from the lane match up, or assistance or level one fights in the enemies favour. Whenever you do not see an opportunity to gank or have to put a lot of pressure on a certain lane, you should take the opportunity to farm your jungle or go to base and buy if you have money for a big buy (ie. getting Bonetooth Necklace). Look at Tactics section for how to gank efficiently. Counter jungling is a decent choice for Rengar since he has decent crowd control and fighting ability(read the Tactics section for counter jungling info). In addition, if one of your laners require you to hold the lane while they back, do so as well. You also want to control both your blue and red buffs, as well as the enemies. Timing the respawn time or knowing the general respawn time of these buffs are crucial for doing so. Control of Dragon also becomes a big factor and whenever your team has a safe opportunity to do so, you should take it. Rengar is not a jungler I would like Oracles on since he dives in team fights and is not particularly mobile cross map.

Mid Game

The mid game is quite similar to the early game. However at this point, you are farming less as your carries may be taking your farm, and you have to be in position to gank and contest Red and Blue buffs, Dragon, and Baron. The same things you do to control those, applies in the mid game, time the respawn of the neutral objectives and buy wards to make the job safer. The power of Oracles persist in the mid game, however you want your supports to get it. Pink wards can suffice for the sake of securing Dragon. You are still able to gank well due to the ability to by pass wards or walk into lane into leap range with your ultimate, and you provide a lot of damage and a bit of crowd control.

Late Game

In the late game, you are an assassin, you have to be killing or at least be a huge disturbance to carries. Itemize defense for the carry you want to kill, killing Brand? get Hexdrinker and Spirit Visage, killing Ashe? get Randuin's Omen. Baron also becomes the focal point of the game, with most fights taking place around the area. Basically, group with your team and get control of Baron. Have your support get an Oracles and make plays with it.

Tactics

Team Fighting

In the team fight, you want to dive the enemy carries or focus off a crowd controlled target that is out of position. By using Thrill of the Hunt, you are able to get extremely good positioning to leap onto and destroy the enemy carry. As long as the enemy does not have Oracle's or a pink ward, you have 7 seconds(or 10 seconds if you somehow get 14 stacks) to wait for a good opportunity to jump the carry. Good opportunities are after they have used major skills, such as escapes or crowd control skills. Make sure you are at full ferocity before leaping else your damage will be sub par.

An example is you are trying to dive an AD, in this case Ezreal. He just used Arcane Shift to reposition, so he you have a time window to jump onto him without being able to get away by himself without using Flash.

Counter Jungling

Rengar at killing a single target quickly, making him a decent counter jungler. However, he has practically zero ability to escape after getting caught. The only ability to helps to him escape well is his ultimate, and that is not a skill you want to use for the sake of a Blue Wraith. When counter jungling, you must be aware of where the enemy jungler was last seen, and the status of your lanes. If creep waves is pushed towards your side, it is more likely you can be caught in the enemy jungle. You should almost never attempt to counter jungle if you do not have an idea where the enemy jungler is and if there is a ward where you walk through to get into their jungle. 

Tips and Tricks

Ganking Efficiently

It is important to get full ferocity before coming to gank. This will allow you to easily open up with an Empowered Bola Strike to snare and slow your target. To initiate the gank, stand in a bush and leap on the enemy champion if you are able to. However, if that is not a possibility, leap on an enemy creep closest to the enemy champion. Immediately after, use Empowered Bola Strike on the enemy champion. Walk up and follow this sequence:

Auto Attack -> Savagery -> Battle Roar -> Continue auto attacking. Once the slow from Empowered Bola Strike expires (1.5s after the root ends) use Bola Strike to slow them again. If they are getting really close to a tower or Flash away, you can use Bola Strike slightly before the 1.5s, as Bola Strike is a diminishing slow.

Once you have Thrill of the Hunt, you can use the invisibility to walk through wards in the river and leap onto the enemy and follow the above sequence. If you have full ferocity and are able to leap from the bush with your ultimate, you can use Thrill of the Hunt to leap onto an escaping enemy with full ferocity. Remember there is a fade time for Thrill of the Hunt, and the ferocity generation ends if you cancel stealth before you regain ferocity.

Jungling Efficiently

Throughout the early levels, you want to be using your ferocity on Battle Roar to heal your HP. It also deals AoE damage so it does not slow your clear time by very much relative to using an Empowered Savagery in the early levels. Besides that, there is not much to optimize with Rengar's jungling, obviously use Savagery immediately after an auto attack as it is an auto attack resetter, and jump to jungle minions when you can to slightly make you faster.

Synergies and Counters

Synergies

Other champions that play similar and dive on the enemy carries work well with Rengar. Examples of such are Irelia, Diana, Ahri, Hecarim, etc. Gragas also works well with Rengar as Explosive Cask will deal a lot on damage to carries and separate the enemies from the aid of their allies allowing you to easily pick them off. Rengar's ultimate can be utilized extremely well with Orianna to use as an incredible initate by jumping into the enemy team in stealth and having Orianna use her ultimate.

Counters

Shen can be a nuisance to Rengar landing Shadow Dash on you as you are fading, or have faded from Thrill of the Hunt will force you to auto attack breaking your stealth. Twisted Fate's Destiny also proposes a problem in team fights as your invisibility will be useless, however you are still able to leap and gain ferocity. If  Rengar ults, the ideal target he would go would be the carries. Best to do is have your Carries next to each other and have a Lee sin near, and E near them to reveal Rengar and slow his MS and AS. Also he can ult rengar away as soon has he jumps on a carry. In addition, by having a very tanky team minimizes the potential of Rengar significantly. This can be achieved by having very tanky carries such as Urgot or Ryze.

Obviously an enemy Rengar will be annoying as his ultimate counters to an extent yours similar to Twisted Fate, but that won't happen in a ranked game.