I've noticed a lot of people have lost sight of the true goal of League of Legends: having fun. For whatever reason, people either think they have to win to have fun, or they just have to win no matter what. You don't need to win to have fun, as I clearly found out last night.
First thing you need to do is gather up four friends that also play, and queue as a group. The tips I'm giving work best in ARAM, but also work in other game modes. Once you find a game, add a goal to it. For instance, we decided that we would ignite the enemy Syndra no matter what in one of our games.
We still tried to win, but instead of focusing on winning we focused on our own goal. While we still lost, it was the most fun I've had in a while. If you feel you're getting angry at losing all the time, try setting a goal. Rammus and Hecarim are really fun for doing crazy move speed builds on.
Here's another example of a fun game that I found online.
Don't let the have-to-win mindset ruin your enjoyment of the game. Play for enjoyment, you might actually win more games as well. Good luck and have fun!