Obviously anyone can win with any hero but there are definitely easier ways than others. Awhile ago assassins dominated soloqueue like Akali, Talon, and Kassadin. Now that they are nerfed what are the new easiest champs to carry games with?
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The easiest basic answer is win your lane, get your team to group and do objectives, and ward a lot.
It's not really about carrying or playing a specific champ it has alot to do with luck on who you are paired with and how willing everyone is to work together as a team. I know that is fairly vague but seeing what other people are picking at the start and knowing what would work well with that and communicating properly from start of team select til the enemy nexus is destroyed is easiest way to win soloqueue.
For all the communication you want your jungler helping lanes that need help, map control from all players and good farm on those that need it.
Wards are huge if you know your jungler is getting countered hard do something about it don't let him get behind. Anything the enemy team is getting away with is because you are letting them. Punish the enemy team for everything they do if they take a turret take 2 turrets. If you see them split pushing force a fight you know you can win and take another turret down.
Mid players gank bot help the ad get ahead give him assists and kills team gold is great and shows good coordination.
Top players just farm push and communicate what you are doing.
Bot lane farm farm harass farm farm kill dragon farm and harass. Keep wards down to avoid getting ganked.
These aren't foolproof things but you'll win more than not if you end up doing these things. You may not get paired up with people that are going to communicate well or want to communicate at all but if you make an effort at it that's a step in the right direction to easily win solo queue.