First I find out what they enjoy copy and pasting this from another post
First think
Do I enjoy ...
Doing consistent damage?
Taking it for the team?
Helping from afar through heals, cc, and buffs
Nuking (doing "tons of damage") one player
AoE Nuke
Combine any of those you like for instance taking damage and doing damage would probably be tanky ad/ap (rumble,garen,udyr,etc) generally for top and jungle.
After you find the role you enjoy, work hard on it because it's better and more productive to work on what you enjoy, rather than trying to find what you're best at. In the end it's a game, and winning is fun but it's more fun to play what you find enjoyable (winning is a bonus too)
As stated people will learn better and faster if they do what they like to do.
Then like rawrful said I'll take them in a 1v1 with their preferred roles, unless it's jungle. For the first death 1v1, play to win, however, on a VOIP (like skype) tell them what you're doing and why you're doing it. Then, afterwards, when you see the chance go in for the kill but tell them you will, if they avoid it, good for them, if they don't, it's another lesson. After the first death tell them what they did wrong, how they can avoid it, then teach them the fundamentals (farming, harassing, and map awareness).
If they're really new to the game then also give them advice on how to vrs champions with unknown skills. It's impossible to expect someone to remember over 100 champions and their skills. Unfortunately for new players, not knowing enemy champions is a huge disadvantage that they can't bridge with just a day or even a month of training. Tell them to hone their observation skills, do they use skill shots? whats the range? did they induce any effects? Most champions only have 4 spells and a passive, tell them to remember what each skill does each time their opponent uses it. Afterwards, if they had a hard time, tell them to look up champ spotlights and such.
Finally, nothing beats good ol fashioned on site experience, get some normal games going with a VOIP and tell them what they should do and praise/note things they do correctly.