Put pressure on your opponents early and force trades, even from level 1 as Draven has to be agressive to pay off.
Always prefer an offensive support (Taric and Nunu work wonders). More defensive support wonts contribute as much to Draven's Gameplay.
During the laning phase, play agressively and harass the opponents on every chance and dont be afraid to chase, as every Q you catch translates into a lot of damage.
This will get you ahead on the lane, with the problem being that you ll be pushing so you will be vunerable to ganks.
On trades (You should always start with a new Q - and try to keep it for 2-3 hits), let your support initiate, then move in with Q and E for some heavy damage and slow so they can't fight back.
During mid game, no matter what you have chosen to build you will do lots of damage. Your lane will be easily won just by throwing Qs on the enemy AD.
On fights though you should try to stay away from enemy bruisers and deal as much damage as possible. Try to use your ultimate to damage their carries.
Catching Q is a high risk high reward move. If you see you can't land many consecutive hits, opt to not try and catch the axes.
Don't risk dying, you staying alive is more important than killing a support or jungler.
Ask for all available red buffs.
Finally, try to focus what your team is. Your ultimate will be doing ~1000 damage, use it wisely.
Tips & Tricks
Your Q applies a DoT on the enemy so carefull with turret aggro.
Also your Axe bounces towards the location you face. So if you hit an axe and turn around, the axe will land behind you.
You can have up to two axes active at the same time.
Υour ultimate is global and can be used for stealing buffs and objectives, sniping low health enemies who are recalling and farming large creepwaves without having to be there.
Supports Draven prefers:
Tier1: Taric, Nunu
Soraka doesn't fit Draven's playstyle.
What Draven is good against:
Vayne, Ashe, Sivir, AD Kennen, Tristana, Varus, Miss Fortune.
What is good against Draven:
Corki, Graves, Caitlyn.
What Draven hates:
Blitzcrank, Leona, Ezreal. They can see where your Q will land so they can aim their skillshots with great accuracy.