How can I create my own private server for a game like LoL?

The closest you'll be able to get to this is to create a custom game in the client where your teammates and the other team are all bots (or it's just you by yourself). You can't host your own LoL server as all legitimate LoL servers are owned by Riot.

To create a custom game on the Riot servers:

1. Click the big red Play button.
2. Click "custom" in the list of options on the left.
3. Click "Join/Create"
4. Click on the "Create Game" button in the lower right.
5. Configure the game settings, then click "Create Game".
6. If you want bots on either side, click the "Add Bot" buttons.
7. Click "Start Game".

If you play a custom, you can end it at any time without incurring a "leave" penalty.