Early Game try to harrass the opponents as much as possible.
If u see them trying to poke ur Support take the initiative and deal as much dmg as possible on their ADc.
Try to ALLWAYS catch ur Axes since it adds up to ur Passive for more gold income.
At lvl 6 u can allways ALLWAYS get FB on the enemy team if not sooner (ganks,catches,etc).

Use ur High DMG early and get an Advantage to Snowball the enemys.
Once u start ur Snowballing u can simply Use ur W skill to Chase them away( Harrassing them hard) to Catch up to them and get some kills ( if u have Tresh / Annie / BC / w/e just something with a STUN / Hook u can easily 1HIT the enemy Team with a good Combination (Q -> W -> R ( exactly at the same time use ur E to slow them and keep them from gettin away) -> W -> Q and Autoattack all the time in between.

Do not forget to CATCH UR AXES ALL THE TIME!
Axes = Draven -> DRAVEN -> AXES -> AXES = DMG -> A LOT DMG -> Use ur dmg and remember the Axes allways fly the same pattern. Simply try the Axes routes in a custom game its pretty easy to allways catch them.

Mid Game Try to allways get those Drakes if possible u can also Duo them with ur support if the enemy Bot recalled and thheir / your jungler are @ Top.

Allways Use ur W in between ur Cds - Usually i have no Downtime of W since i Spamm it as Soon as it gets rdy and since it allways refreshes CD when u catch an Axe its ur most Used SKILL ^^.

Try to Stand behind ur Team and letting them Peel for you. Dont play passive on Bot ofc its good if u can Farm up but u have insanely Dmg so use that Dmg and simply melt the enemy team in Teamfights, DO NOT FORGET THE FOCUSS u can also disengage nicely with ur E so use ur Abilitys.

Dravens Natural Enemy is simply CC.
If u play against a Hard CC team simply rush Banshee after BT -> LW -> T2 Boots and position urself safely -> w8 till they use the most threating ccs and then go into the fight and melt their Carrys.

Greetings Chaos