Draven is the highest risk / reward ADC in the game. His damage potential is insane and can literally 1 shot carries with his Q if you get ahead of them.
If Draven gets ahead early, he snowball the entire game very easily.

Use standard ADC runes for Draven. Some people swap out the ATS quints for extra damage, but the attack speed feels a lot nicer and gets you to the stage where you can easily throw and catch both spinning axes without stopping.

Again, standard ADC masteries. Make sure you have Frenzy as you will crit a lot once you have your items.

The B.F sword and brutalizer are both early powerspike items. When you have them both (and not noticeably behind the enemy ADC), you will win a 1v1. The flat Armour Pen is great v low armour targets, so you will hurt them more than they will hurt you.
The rest of the build is fairly standard. Furor enchanted boots are important for Draven, the extra movespeed will help massively when catching axes in a team fight and will increase your DPS by a lot by being able to catch them. 2 or 3 axes onto a carry will kill them once you have the Infinity Edge, so being able to use just 4 or 5 axes on the right targets will win a fight for you.

Q - Dravens Q is where his insane damage comes from. The bonus damage on hit is ridiculous late game. Remember to stack this at all times in the game. Catching an axe will also reset the cooldown for your W, so it increases your attack speed and movespeed, making it easier to catch the next axe.

NOTE: Once you throw an axe, the direction in which you move immediately after, is the direction the axe will land. If you throw one and walk toward the enemy, the axe will land nearer to them, which is great for chasing. If you throw one then immediately walk backwards, it will land safely behind you, which is better for landing single hits on champions and for hitting creeps.

Pros:
More damage than any other ADC in the game.
Huge snowball potential due to gold from passive.
You can punish mistakes by using E, W, Q on anyone who is overextending.
The most fun ADC to play.


Cons:
You don't have a dash to escape with.
Junglers should camp you and shut you down early to stop you cashing in on stacks.
If you play badly, everyone will question why you picked Draven ahead of a safer ADC.

This is the hardest part of playing Draven for most people.

In order to maximise your potential gold and then damage, you need to stack as much as possible in the early game so you can then get a kill and earn bonus gold.
While you only have a dorans blade, every minion can be worth 4 stacks, so a wave can be potentially 24 stacks (will probably get less due to minion focus meaning some are already low). Getting these early stacks means you can gain a huge early gold lead with a single kill.

Every stack is worth 2 gold and then 50 bonus gold on top of it. so with 250 stacks, you will get 550 gold [ (250x2)+50 ]. Once you add the kill gold on top of that (usually over 300g), you're massively ahead in terms of items once you go back to base, and you are much stronger for the next fight, meaning you can win it easier and snowball from there.

Controlling the axe landing area is often overlooked when it comes to new Draven players, but it can lead to a lot more axes caught and a lot more Q's being thrown at enemies.
By simply clicking a direction while the axes is heading towards its target, the axe will head in that general direction. If you stay still, it will land either on or very close to you.
This means when you are chasing an enemy, the axe will be ahead of you, so you can keep running, catch the axe, reset W cooldown, throw another Q, run after it with increased movespeed from W, then repeat until fed.