Soldiers don't do too well on it, its movements are too erratic to reliably hit it. I hear the Reserve Shooter always minicrits it behind the scenes without the visible minicrit effect, so that is a good thing to use due to no dropoff whatsoever.
Scouts do alright if they have the Shortstop. You need to keep a good distance to avoid the crit rockets it constantly spouts out. Eye can't be marked for death.
Pyro is essentially the Soldier entry on the Reserve Shooter. No afterburn on the Eye.
Sniper does alright, sometimes it's hard to get a good line of sight on it. Eye cannot be jarated. DPS is gonna be low due to no crits.
Demo is difficult unless you are a god with grenade spam.
Spy can do alright with a revolver. Those things hit pretty hard and you can use the Dead Ringer to escape sudden aggro.
Heavies as always do the most damage, but they will die pretty quickly as well. Ambushing the eye makes its health drop like a rock.
Engineers take a while to set up and the eye teleports around, so it isn't terribly effective. If it pops up near a level 3 sentry it is only marginally worse than a heavy in terms of health-rock-dropping. Unless you have the Wrangler it will stop the other team from hurting the eye, so it isn't really worth it.
Medics as always are just there for crits.
The strategy for taking on the eye is the same as winning the game: Spam heavies and medics.