In case this answer is a TL;DR, I have highlighted the important parts in bold.
I have had to carry many games when I had teammates who were having trouble.
Often, when teams start to do badly, players begin to throw around blame and get mad at each other. As the leader of the team, make sure you do not blame others and try to help calm everyone down and motivate them to win. (This is hard to do sometimes, I can admit but faking it well works too)
Also, make sure to communicate. Example - Baron/Dragon is up, Enemy respawned, Protect inhib, etc.
I have found that small acts of kindness can motivate your team to win, such as giving them blue or red buff, or letting them get the kill during a chase.
Remember that one fed champion is not enough to carry, because the enemy can easily build items to counter you.
- If someone on your team is feeding, avoid things such as typing /all 'Report "name" for feeding/lack of skill!' This will only demoralize them from trying and will do nothing to help your team.
- Try not to tell them they are feeding. They know they are, and reassuring them about their poor play won't help.
AFK farming is probably not the best choice, especially in higher levels of play. If you have an obvious presence on your team, your team will begin to follow you. Ganking is very useful to your teammates when they are falling behind, but a word of advice-
- Try not to take all the kills. I understand as the carry it is the other player's job to feed you, but this mostly happens only in premade/ranked games. Try to split up the kills, because few solo queue players enter a game without thinking - 'I would like to get fed'
Good luck carrying!