(So you know, this is my first guide. Please tell me anything I can improve on in the comments)
So, you want to learn how to thresh, eh?
Well, you're in luck! I will be splitting this guide up into 5 parts:

Why this build works
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How to play thresh as an Attack Damage Carry
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Early game
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Mid - late game
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Conclusion/Tips and tricks
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I found that the overall ability of ADC Thresh is directly related to how skilled a player is with him. This means if it is your first time with him, YOU WILL MOST LIKELY FEED. Do not get discouraged. It takes a long time to carry as Thresh. Personally, I have been playing him as my main ADC for many months now, and it really has taken that long to dominate with him.

NOTE: I have highlighted important facts in blue.

Why are these Thresh's final items? Most ADC thresh players know that thresh is meant to have heavy AD with a little bit of attack speed, but I clearly have the opposite.

1. Firstly, I should explain the importance of rushing the Statikk Shiv. The Statikk Shiv's passive is amazing on thresh. Its "build up" period syncs very well with your passive effect on Flay. Also, early game, it is ideal for poking, as the shiv damages your foes while you sit back and last hit minions.
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2. The Berserker's Greaves are pretty straight forward, just an additional 20% attack speed. Good on AD carries in general.
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3. Phantom Dancer is what throws many people as my third item, because usually they want damage ASAP. However, these are my reasons why this item is third on the list:

Adds additional critical strike chance (now 50%). your Statikk Shiv can also critically strike. Critical strikes are your laning phase damage dealers.

Gives you plenty of movement speed when combined with your Statikk Shiv and Berserker's Greaves. This insures you to catch up with running foes and drag their faces back with your Death Sentence.

The high attack speed is good for farming, and the more minions you kill, the more souls you can pick up. This is VERY important because you get most of your souls during your laning phase. Not only do these souls give you armor, MR, and AP, but every single one deals +1 damage with the passive on Flay.
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4. Blade of the Ruined King is amazing. Not only does it give 25 AD, 40% attack speed, and 15% lifesteal, but its passive deals 5% of the target's current health.

For example, if your target has 3,000 health , the Blade of the Ruined King will deal 150 bonus physical damage (negating other effects). Its active will deal 450 damage each hit.

This item doesn't even have to work with thresh. It's just an amazing item that every ADC should have.
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5. Bloodthirster is your first heavy AD item. It has the highest amount of damage of any item in League of Legends (other than a fully stacked Sword of the Occult). Little explanation is needed.
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6. The last and probably most important item of your entire build is the Infinity Edge. It's passive (grants an additional 50% damage onto critical strikes), 70 AD, and 25% critical strike chance screams Thresh. After this item you will have 75% critical strike chance and deal massive damage.
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FIRST HIT ON ENEMY WITH 3,000 HEALTH MATH:
257.35 x 2.5504 = 656.34544
AD.......Crit
257.35 x 2.0 = 514.7
AD......Flay
100 x 2.5504 = 255.04
Shiv..Crit
(avg. 150 souls) = 150
BRK passive = 150

Add them all together to get 1726.08544 damage (not including armor or MR)!

Thresh is no ordinary ADC (obviously). His position in lane should be passive, staying behind minions, and moving to a new position that is good for hooking after each auto attack. This way, your opponents will become familiar with your actions and believe that they are safe, when in reality you are attentively waiting for your moment to strike.

Try to grab the other ADC if their support is too tanky. Thresh is good at melting squishy characters. Also, don't throw your hook out too much. This will burn mana quickly and reveal where/when you plan on hooking them in the future. And, most importantly,land your hooks. this is extremely necessary in order to get free damage off.

Once you've landed a hook on your victim, you have 2 options. Your first is to thwak them with an auto attack/auto attacks and back off. Your second choice is to leap onto your victim, but it is a risky act to do. Consider these circumstances before you press Q again to jump to your victim:

1. Will I take too much damage/die?

2. If I jump in, will I most likely kill my victim?

3. Is the jump destination under a tower that I should not dive?

If you do go in, make sure that you prepare your Flay in mid-air. Cast it away from their tower once you are on top of them, slowing and knocking them back into your clutches. Also, throwing your lantern down is a good idea to absorb any damage you might take.

Make sure your support knows how to use your lantern. In some situations, it will save their life.

Speaking of which, Thresh works well with aggressive supports that have high CC such as

Leona, Blitzcrank, Janna, and sometimes Soraka.

There aren't too many ADC counters to thresh, but there are supports that shut him down. Blitzcrank and Morgana can easily shut him down if played correctly, so watch out for them.

Thresh is pretty weak early game, and you should play passively until you reach level 3. However, he is very good in first blood - bush fights if he can initiate with his death sentence.

Collect souls for extra damage and build up your Flay passive before poking with an auto attack. Also, as explained before, landing hooks is critical for dishing out damage and sealing kills. Your item goal is to have your Statikk Shiv, Berserker's Greaves, and Phantom Dancer by the end of the laning phase. If the laning phase ends early, swap out phantom dancer for a Blade of the Ruined King and get back to it later.

Thresh is comfortable on his tower because of his hard CC. One hook can mean instant death. Plus, since he can turn on a situation rather well, if anyone is stupid enough to dive you, throw down your lantern for a "hidden" amount of health and CC the hell out of them until the tower wrecks them.

Toss out you lantern instead of face checking a bush. It will save your life many times over. trust me. Your lantern can also be used for super - ganking by throwing it to your waiting jungler. For more Tips and Tricks for Thresh, take a look at his champion spotlight, located here.

NOTE: This is the spotlight for a support thresh. Some rules may not apply.

Now that you've made it this far without the enemy team surrendering, you need to be able to carry the game!

Because Thresh is item dependent, he is considered a "late game" champion. This means that he will deal tons of damage, and he can murk pretty well, too. (to murk: to deal a substantial amount of damage in teamfights even after losing lane)

DO NOT JUMP DIRECTLY INTO THE FIGHT! Many, Many, Many, Many, Many, people make this mistake, and they pay for it with their life almost any time. It only makes sense: if the enemy team focuses the carry and the carry jumps into their face, then the carry is pretty screwed. In team fights, try to position yourself away from the frontlines. Let your tank go in first, then follow up with your strong auto attacks. Use The Box when it is necessary to escape or when a majority of the other team will be hit.

Your Death Sentence can be used as an escape tactic, too. Similar to Lee Sin's Sonic Wave, you can use it to latch onto neutral monsters or minions and fling yourself onto them. Careful - Thresh is squishy - at level 18 he has only a little more than 2,000 health. If someone is chasing you, Flay them away to stun them and slow them for a short period of time, allowing your escape. your Death Sentence will also render them unable to move for a short amount of time, so keep that in mind.

Thresh works best with another champion by his side because they can land their attacks while he uses his CC (this doesn't mean that he can't go alone). When it reaches the time to grab objectives, try to aid your team in capturing them when you aren't farming lanes. You can save their lives if an unfavorable matchup attacks them in the jungle. Know that Thresh deals much more damage to champions and creeps as opposed to towers. Flay's passive and critical strikes do not apply to them.

Overall, Thresh is a unique carry who deals more damage as the game advances because he is item dependent. Limit the use of Death Sentence during your laning phase, as you will need the element of surprise.

Flay's passive damage buildup makes up most of his damage. Hold off your auto attack for a few seconds then land it on a champion to take a hunk from their health.

Playing ADC Thresh correctly will take lots of practice, so don't think he's a horrible carry/you are horrible with him if you lose your first several games. Don't forget to rate my guide and comment, it was my first!

A few tips:

Plan your hooks according to Thresh's wind up time. It's pretty lengthy.

Never collect a soul if you will take too much damage trying.

Use your lantern as a ward to check things like Baron and Dragon.

Flay people into the walls of The Box.

You can hook/jump on to people through walls.

Thresh can kite fairly well at a closer range.

Have fun!