To remove one of your game accounts, you go to the left sidebar navigation called Your Games.
Hover your mouse over the game you would like to change/remove your Game Account for. Then click the three dots (...) that appear on the right side.
After clicking the dots a menubar will appear, click Edit Steam Account at the bottom for any game that is played via the Steam platform.
For League of Legends the text would say Edit Summoner, as on the example image below.
You can also do this by going to the Game Overview Page.
You get there by clicking the game title in the Your Games menu. When you have entered the game overview page, you will see your Game Account appear below the game title with a cogwheel Icon next to it.
This button has the same function as the Edit Steam Account/Summoner button has.
Please note: If you have any kind of ongoing activity, even if you are only in the team finder and no tournament or ladder, you cannot remove your game account.
Double check that you don't have any ongoing activities before trying to remove a game account.
Remove the account
Click on REMOVE and your current account will be disconnected from your Challengermode account.
After the account has been removed you can add a new one.
Note: You can only have one account for each game. For League of Legends you can have one account connected for each Region.
Add a new account
If you have removed the account and want to add a new one, refer to these articles. They will take you through the process of adding a new account for that specific game.