League of Legends Time Played: How to Calculate
Unfortunately, or maybe not, Riot Games has not implemented a way to check how many games you have played into the League of Legends client just yet.
Even the third party resources for this topic are not completely accurate due to some difficulties with Riot's API and name changes.
League of Legends Time Played: How to Calculate
The website with the most generally accepted results is here. Even then, the website will not be completely accurate if you have switched either your Summoner's Name or the region you play on.
Additionally you can look at your wins and losses after a game to determine how many games of each game mode you have played. Once you have determined that number, you can multiply it by the average length of each game mode, which can be around 25-30 minutes for each game.
Both of these methods are going to be rough estimates so do not put a large amount of emotion into the number you get whether it is large or small.
Cover Image Courtesy of Riot Games