How do you send a personal message to another player through ingame chat? or out of game chat?
In game you can type:
/w "[player name]" [message]or
/msg "[player name]" [message]. After they've messaged you once, you can just type:
/r [message] and it'll go to whoever messaged you last.
In lobby/champion select, you can double click on their name to bring up a chat.
Note: this only works with the ones in your friends list.
There is another way to do this should you not be able to type their name due to special character or simply forgot.
You can alt tab click on the tray icon and click on the league icon as seen here:
You can then just use the normal chat in this window. Double click there names and go!
Another benefit of this is it will store the messages you send not just the ones you receive (Which happens when using /w)