Carrying as an ADC can be tough in solo queue, you are a huge source of damage but very squishy, the enemy team will focus you and without a frontline or protection you may just die every team fight. Carrying as a support might actually be easier if you focus on helping your ADC out every team fight to make sure his DPS can go on.
Now, if you are set on carrying as an ADC i think you should play the role for what it's meant to do, pushing towers. The only reason an ADC is played in every ranked game and LCS game is because you need one to take down turrets faster. You can give your team a huge advantage by following these steps:
Step 1. Get ahead in lane, either by CS or kills.
Step 2. Take down the turret and push up the lane.
Step 3. Rotate mid and top and take down those turrets.
Step 4. Push up side lanes and group mid to push as 5.
This is what an ADC excels at, the first 3 steps can be fairly easy to complete, the hard part might be grouping with your team. Taking down the first 3 turrets gives your team a huge advantage and team fighting at this point should be in your favor because of the global tower gold. A lot of times, games are lost simply because one lane lost super duper hard, if you take down the turret in that lane and start to group instead, your team mate who previously fed won't just die over and over until it's too late. My point is, every role has something that helps the team, could be a kit with CC, roaming, ganking, warding, buffs etc, the ADC has pushing power, use it to help the team out, just like a mid laner should roam, just like a top laner who tps, or splitpushes, just like a jungler who's ganking.
Do all this, or just play Draven and snowball like crazy and become totally unstoppable.