Aphelios needs to be toned WAY DOWN
The guy simply does too many things and does all of them fairly well. One big issue is that most of his power comes from the passive aspects of his kit rather then something easy to fix like high ratios. Another is that since his kit auto levels, and thus increases all 5 of his abilities at the same time, he naturally has a stronger early game then all other ADC's. This might be 'ok' if he fell off in the late game, but of course we can't have a carry champion do that.
I suggest starting with some of his "invisible" power:
Nerf his poke, something simple like reducing the vision time by 1.5-2 seconds and changing the bonus range from global to just a large extended auto range. Like 
traps/net give.
His sustain should probably be looked into. Even if he doesn't always have access to it, a passive 8-25% lifesteal is VERY strong.
Much like his sniper rifle, i would suggest some minor nerf to his gravity gun. I would either reduce the slow amount and/or the duration of that slow. Then i would put a range limit on his ability to trigger his root. It needs not be tiny, but again global range on a core kit ability is just too much.
Now we get to the two biggest problem weapons. I have many issues with the flamethrower, First is that it gives a rather large bonus range with it's passive cone effect. Next we might address the fact that this weapon gives a passive cleave (a feat i don't think any other champion has). Then we can address the fact that the cone does 75-100% damage to secondary enemy champions hit, meaning that even if he is targeting minions he does impressive damage to enemy champions who walk towards him to engage. Finally i would address the active ability for this weapon, specifically the fact that Aphelios gets a free auto attack (again at global range) on ALL targets he hits, which also triggers the off hand passive of the weapon used. This means that this weapon will generally have somewhere around 200-1125% bonus attack damage in the form of free auto attacks. The thing to keep in mind here though is the off hand passive, not only will you get a free auto attack but that auto attack will do 1 of 4 things. It will mark the target again allowing for yet ANOTHER free auto attack, it will heal Aphelios for 7+lvl% per target hit, it will slow all those targets by 30% for the next 3.5 seconds (and let you gun swap for a root if you wish), or it will spawn somewhere between 2-5 (possibly more) mini chakrams.
Lastly let us address the train wreck that his his Chakram. I do not believe he can be balanced with a weapon that has a passive auto attack reset. We have seen how insanely powerful animation canceling/auto attack resets are on other champions. This weapon can remain very fast, but he can not remain able to break max attack speed, much less do so with as little as one item.
Finally i would address his turret. This thing is far to strong. Why does the turret need to benefit from his attack speed and crit? Why does a 'pet' need the ability to crit? With the turrets 550 range, most champions (even ranged ones) can not get close enough to auto attack the thing without being within its range. The turret triggers the passive of what ever gun Aphelios used to set it; so it's either giving him global range vision and auto attacks, healing him, attacking with a strong passive AOE cone, or slowing everyone hit (that he can turn into a root from global range). Let us also not forget that the turret can not be targeted until it's active, so even if you do know it's there (Aphelios can also leave multiple turrets if he has some set up time) and you do have the range to kill it without it attacking you, you are prevented.
