League of Legends (LoL) Question: What champions are good for going up in ranked ( ADC ) ?

I'd say Jinx, Tristana, and Xayah. They all have some kind of self-peel and scale extremely well. Stay away from difficult champions like Vayne and Draven. They are not rewarding for the amount of effort it takes to play them. The most important thing you can do to climb is stay safe and farm well, losing early game will ruin your chance of scaling up and winning. Make sure to ward and play around your jungler.

Stay away from ADCs like Vayne, due to her skill floor. Although I'd say solid picks would be Jinx due to her ludicrous damage after two items AND her delicious resets. Tristana because of her late game damage and her survivability. Kai'Sa would be a good pick if you are willing to invest time. In low elo, Twitch would be a really solid pick as well because of his damage after a few items, and invisibility. Of course, I cannot leave out Xayah. She has amazing waveclear, CC, survivability, and pushing power. Even better if you can find a Rakan duo.

Just remember, destroying the enemy NEXUS is always the name of the game.

I recommend trying out Tristana and Jinx. Tristana is decent at all points in the game while being really safe with Rocket Jump and Buster Shot. Jinx is strong right now and is one of the best lategame hypercarries. With low elo games tending to last longer, she's really useful.
But in fact you can climb with any champion. Keep on learning things while having fun and you'll climb automatically. You can check out Vapora's great Tristana guide for both champion specific and general ADC info if you want.

If you're experienced enough, Vayne. I strongly recommend you main her if you're not, and learn how to farm in difficult matchups. Master that, you're able to solo carry regardless of your team.
and of course make sure you actually go for objectives.
last tip: Don't feel obliged to go in whenever your support does.