This guide isn't a complete guide. I've directed this guide more to Lee Sin's abilities and how he can use his abilities together. I decided not to include the build, to keep this guide more on-topic.
This guide will teach you how to play Lee Sin effectively.

Pros

+High Damage Output
+Great Mobility
+Doesn't use mana (Not reliant on buffs)
+Extremely strong early game
+Great ganks
+Fun To Play

Cons

-High Skill Cap and Learning Curve
-Tends to fall off late game
-Needs to land skillshots

- Greater Mark of Attack Damage provides great early-game damage for our ganks and jungle clear. Standard choice.
- Greater Seal of Armor provides much-needed defense against jungle creeps and reduces incoming damage from opponents. Standard choice.
- Greater Glyph of Magic Resist gives us flat magic resistance, which helps negate early game damage from enemy AP champions.
- Greater Quintessence of Attack Damage provides more flat damage, giving us a total of 15 attack damage just from runes, which is no joke on Lee Sin.

Alternative Runes

- Greater Glyph of Scaling Magic Resist is a good choice if their mid laner is AD, or if you prefer it in general.
- Greater Quintessence of Life Steal can help you sustain more in the jungle. Combined with Iron Will, this is a solid choice on Lee Sin. If you're taking these, take 1 Greater Quintessence of Attack Damage and 2 of these.

Smite is a must have on all junglers. It helps in your early clear, and secures objectives like Dragon and Baron Nashor.

Flash has plenty of uses, helps to escape, chase, reposition and more. Can be used to get behind enemy champions and Dragon's Rage them back into your team. More explained later.

Ignite can replace Flash if you're confident in your ability to ward-jump (explained later) and want the extra damage to secure kills. Helpful against champions like Dr. Mundo.

Passive

Flurry
After using an ability, Lee Sin's next 2 basic attacks within 3 seconds gain 40% attack speed and return 15 energy each.
Explanation:

Q ability

Sonic Wave/ Resonating Strike
Sonic Wave : ACTIVE: Lee Sin shoots a wave of sound in a line, dealing physical damage to the first enemy it hits and granting true sight of it for 3 seconds. If Sonic Wave hits, Lee Sin can cast Resonating Strike within the next 3 seconds.
Resonating Strike : ACTIVE: Lee Sin dashes to the enemy marked by Sonic Wave, dealing physical damage including 8% of the target's missing health. Minions and monsters take a maximum of 400 damage.
Explanation:

W ability

Safeguard/ Iron Will
Safeguard : ACTIVE: Lee Sin dashes to target ally, if the ally is a champion, he and the target each gaining a shield that absorbs a certain amount of damage. Each shield dissipates after 5 seconds if not already destroyed. Lee Sin can cast Iron Will within 3 seconds of using Safeguard.
Iron Will : ACTIVE: For 5 seconds, Lee Sin gains bonus life steal and spell vamp.
Explanation:

E ability

Tempest/ Cripple
Tempest : ACTIVE: Lee Sin smashes the ground, dealing magic damage to all nearby enemies and granting true sight of them for 4 seconds. If Tempest hits an enemy, Lee Sin can cast Cripple within the next 3 seconds.
Cripple : ACTIVE: Lee Sin cripples all enemies marked by Tempest, reducing their movement and attack speed. The movement and attack speed recover over 4 seconds.
Explanation:

Ultimate

Dragon's Rage
ACTIVE: Lee Sin launches a powerful roundhouse kick, dealing physical damage to a target champion and knocking it back 1200 units over 1 second. Enemies the target collides with take the same damage and are knocked airborne for 1 second.
Explanation:

I've seen countless Lee Sin players who don't know how to effectively use their abilities. In this section, I'm going to be highlighting the key combos a good Lee Sin player will be able to execute.

IMPORTANT - Smartcasting wards. If you don't like smartcasting wards, at least change it to smartcast when playing as Lee Sin. Smartcasting wards makes it possible to Ward-Jump (explained later) efficiently, there isn't a way to do it well if you don't smartcast wards.

Combos :

+Ward Jumping

->
Lee Sin is known for this technique. It's the act of dropping a ward, usually over a wall, and using Safeguard to it. A well-executed Ward-Jump can effectively act like a Flash.

Poke

-> ->
The act of hitting Sonic Wave, then activating Resonating Strike and immediately using Safeguard to an allied minion/champion to avoid retaliation.

Execution

-> ->
The act of hitting Sonic Wave, then using Dragon's Rage, and then activating Resonating Strike to get to the enemy. Using Dragon's Rage before Resonating Strike maximises the execution damage from Resonating Strike.

Kick their carry

The act of using Dragon's Rage to hit an enemy champion, usually a carry, into your own team. There are a couple of ways to do this :

Beginner's method

-> -> ->
Using Sonic Wave and then immediately activating Resonating Strike to get to the target, then using Flash to get behind the target, and followed by Dragon's Rage to kick the enemy into your own team.

No Flash/Ward Method

-> -> -> -> ->
In the unlikely occasion where you don't have either Flash or a Stealth Ward, you can do this by hitting them with Sonic Wave, immediately activating Resonating Strike and using Tempest, immediately followed by Cripple for the slow, then use the slow to walk behind them, and use Dragon's Rage to kick them back. Obviously you aren't going to be doing this in teamfights, usually this is used for ganking.

No Flash method

-> -> -> ->
Using Sonic Wave and then immediately activating Resonating Strike to get to the target, then placing a Stealth Ward behind the enemy champion, followed by a Safeguard to the Stealth Ward, then using Dragon's Rage to kick the enemy into your team.

Ultimate Gap closer

-> -> -> -> ->
If you're really far away from their enemy carry or behind a wall, this will help you unexpectedly close the gap between you and the enemy carry.

Smite Heal

->
This will help you heal off Smite, keeping you healthy in the jungle.

Ultimately, you have to get in game and practice these combos. You're not going to become a great Lee Sin player just by reading this. Keep practicing, you'll get better at him.

The best jungling route, in my opinion is to mirror the enemy jungler. This means that if the opposing jungler starts blue, you should start red. This can lead to very deadly counterganks and safe jungling.
Alternatively, Lee Sin usually starts at the blue buff, then does wolves, then heads over to either the wraiths or the red buff.

Lee Sin is an extremely strong duelist with his high early game damage output, his movement and attack speed slow from Cripple, and his high mobility. If you're up against a champion that's prone to counter-jungling, such as Amumu, make the best out of your early-game by not only ganking, but also counter-jungling.
For example, it's easy to get to Amumu at his red buff and easily kill him or at the very least, force a flash. Shutting buff-reliant champions down by taking their buffs is a very effective strategy. Lee Sin doesn't have the best clear speed, compared to the likes of Shyvana, but is certainly decent when it comes to clearing. Taking the opposing jungler's camps is a great way to stay ahead as Lee Sin. Shutting down an enemy jungler will release pressure from your lanes.

Although Lee Sin is indeed a strong ganker, ganks can often fail due to mistakes or not knowing how to properly execute a gank.
There are a few primary ways to start a gank.

1) -> -> -> ->AA/
This is a effective gank technique. You run in from the side, Safeguard to an enemy melee creep who's preferably in the middle of the creep wave, then immediately double tapping E for the slow from Cripple. While the enemy is slowed, Sonic Wave will be practically impossible to dodge due to the fact that you're probably at point-blank range and that he's slowed. Enemies will usually try to Flash at this point. If they didn't use Flash, you'll most probably be able to land your Sonic Wave. When you land your Sonic Wave and are still in close/melee range, keep hitting auto-attacks to get him as low as possible before using Resonating Strike. If your opponent for some reason panicks and uses Flash when you already tagged him with Sonic Wave, you can easily use Resonating Strike to close the gap. However, if the enemy is being smart, you may be baited into using Resonating Strike, while they flash into tower range, and if you're low, they may just take you out.

2) -> -> ->
This is when you use Sonic Wave to engage a gank. This is immensely harder to pull off, as the enemy won't be slowed from Cripple, and they're able to dodge it. This however should be done if you need to close the gap from you and your opponent and aren't in melee range.

A common mistake some junglers make is that they think that they can't take the kill, and always let their laner try to take it. Sometimes, when they've gotten the enemy low enough that they could take the kill for themself, but tries to let their laner take it, and when the laner gets in range, the opponent Flashes away and they've found out that they've wasted that opportunity. While you should always try to get your laner the kill, you shouldn't be afraid to take it for yourself. Just throw everything at the enemy and do your best to secure the kill, even if it means taking it for yourself.

Another thing worth noting is that if you've manage to burn a flash on a gank, make sure to capitalize on it! A burnt flash doesn't mean anything if you don't gain anything off it. Gank the lane again before the enemy has his/her flash off cooldown.

Post-6 your ganks would be easy to pull off, just follow the steps in the Skill Combos section.

It's strange how a single buff got a whole section of its own, but I've felt that it was necessary. A common misconception is that a jungler must always give their second blue buff to their mid laner. While this will benefit the mid laner, and Lee Sin doesn't need blue buff anyway, it would be better for Lee Sin to take it, rather than a mid laner who takes it, and then gives it to the opposing mid laner a few moments later. The point I'm trying to make is that if your mid lane looks like it's just going to end up feeding the blue buff to the enemy, it'll be better if you just take it yourself.

This section is to share some strategies that I've found to be very useful in gameplay.

-Don't waste time. If you sit in a brush for more than 15 seconds, you could have used that time to farm a wraith camp or a wolf camp. Don't waste time waiting for nothing, if you've been waiting for a long time, you'd be better off trying another time.

-Prioritize ganking winning lanes. Don't gank for a Vayne who's 0-5. The most is that you help your Vayne secure a kill, but considering the fact that she's already died 5 times, it's likely that she'll just end up feeding again. With that time, you could have ganked for your LeBlanc, and extend her 4-0 lead, take the tower and let her snowball. Try to gank lanes that have the potential to snowball.

-Consumables. Make it a point to purchase some consumables every time you back. A Stealth Ward and a Health Potion is good to start off with. This means that you buy this every time you back. Trust me, wards win games.

-WARD, WARD, WARD! Wards win games. Vision wins games, nuff' said.

-Timers. Keeping timers are very, very, very helpful. Keep timers of your buffs, enemy buffs, dragon and baron. Knowing when objectives are up can prove to be very helpful.

Thank you for reading this guide, and I sincerely hope that you've learnt something from this guide. Please leave any constructive feedback in the comments section. With that being said, I wish you all the best in League. Have a nice day!