I've heard of people that try to only play aggressive arena decks, but drafts don't usually allow for the same kind of aggression in my experience. I could be drafting it wrong though.
What do you mean by "the same kind of aggression"? The same as what? Of course it can't be the same as constructed, where you try to empty you hand and kill your opponent in like 5 turns.
The important thing to note is that every deck is weaker than in constructed, so being unable to kill your opponent fast is not a huge deal. He won't automatically have great stuff to shut you down in the late turns. There are far fewer Sludge Belchers, tauning giants, Alexstrazas, etc. Really all arena decks are some sort of midrange. Arena "aggro" is just aggressive midrange.
What you want to do is contest for the board the first 4-6 turns, taking opportunities to hit face instead of trading if there's no real way for your opponent to get better value than the trade. Then start to be more aggressive once you get a tempo advantage and be willing to give up board position if you can deal enough face damage. Like if you each have 8 damage worth of minions, you can hit face instead of trading, and even if your opponent can get some favorable trades, if you have the ability to pressure his life enough, it won't matter. This requires you to have some burn or good removal (to mitigate the board disadvantage). Mage and Hunter are the best for this since they have good AoE (Flamestrike, Multi-shot), single-target removal (Poly, Deadly, Mark), and burn (Fireball, Kill Command, Steady Shot)
When you draft, you don't actually want to draft that differently than you would for a board control deck. A few things like Rocketeer, Nightblade, and Leper Gnome become better, but the top cards are still the same. You're still taking basically every Ogre you get offered.
Recently I've been trying to get my Hunter level up, so I've been playing Hunter as much as possible. In 9 runs in the last couple months, I've gotten at least 5 wins every time (9,5,7,7,6,5,5,8,11). That's a reasonably high floor. The point is that even if you get a bad draft, a lot of arena decks just can't handle the face pressure. When I drafted this deck
0 - hunter's mark
1 - arcane shot, tracking, archer, webspinner
2 - explosive, freezing, mad sci, engi
3 - companion, deadly, kill, 2x unleash, arcane golem, coldlight, farseer, panther, cleric, silverback
4 - 2x multi, dwarf, 2 houndmaster, twilight drake
5 - explosive, faceless, buzzard
6 - ogre
I thought I'd be sub-3 wins, since I lack any real early game, but it still went for 8. Now maybe you could say this is not really an aggro deck, but I think it is. It starts going face relatively early, giving up board position to get saved by the removal while minions go face.