This does not seem like a connection issue, and more like a computer issue.
First thing you should try doing is open Task Manager while in-game, and check if anything is eating up your computer resources, especially during a client freeze, and end its process. You can also run League as borderless instead of fullscreen, making it easier for you to close the game if you experience a problem through Task Manager.
If you are unable to locate/fix the problem with the Task Manager, checking to see if your Graphics Card Driver is up-to-date can solve many issues occurring with your game. Using the automatic tool from your Graphics Card Manufacturer can find and install updates to your driver.
I will provide a list of the common Graphics Card Manufacturers and their tools to update drivers:
If your drivers are up-to-date or if updating your driver does not fix your issue, you may need to run a "Full Repair" from the League Client. This will repair corrupted files that could be causing game issues.
If neither solution works, send a Support Ticket to Riot Support to get more thorough help.