Basically, every time I click to move (or use another key I've bound to move) I head towards the enemy nexus, via the mid lane. This makes the game unplayable. Has this happened to anyone else, and if so, how do I stop it?
This is a Bug that has been reported several times in the League forums. Some of the players claim it's a hack although it is most likely related to some form of connectivity issue.
While I haven't experienced the bug myself, closing and reopening the game seems to solve the problem for all players.
One player got a response to the bug and posted a screenshot of the ticket on Imgur.