It takes one game which is the tutorial game to reach level two, another to make level three. I think at level three you can play PvP but just can't enter the chatrooms until level five. Really unless you are amazing it is worth playing a PvE game or two each time you are learning a new champion. In fact I probably play them 10 times PvE until I really have the feel for them so I don't let my team down and end up getting trolled just for one missed skillshot in a teamfight, "Stfu noob u suk".