I posted this in the Bug Report forum as well but I figured it'd perhaps be good to post it here as well.
As a few people have reported, the client crashes a lot on Windows 8 and I got tired of it so I decided to try and fix it myself. Running Visual Studio's debugger after a crash indicated an issue with Adobe Air. The version of Adobe Air packaged with the client is 3.1 which is from late 2011, while the current version is 3.5. Since Adobe Air is a managed runtime there isn't really any reason not to update it every time it gets updated but it's RIOT's call when they want to update. Anyhow, the PvP.net client will not use an installed version of Adobe Air and will only use the embedded version. The solution I found was basically just to update the embedded version to the newest version and so far I haven't had a single crash. If you want to give it a try and see how it works for you here it goes. Download the installer from http://get.adobe.com/air/ then either install it or extract the package to a folder. If you install the folder you'll be looking for is located at "C:\Program Files (x86)\Common Files\Adobe AIR\Versions\1.0" by default if you're running 64 bit Windows 8. There is no (x86) if you're running 32 bit. Copy everything in this folder to "C:\Riot Games\League of Legends\RADS\projects\lol_air_client\releases\0.0. 0.226\deploy\Adobe AIR\Versions\1.0" or change the path to your install if you didn't install to the default location. This (so far) has fixed my problem 100% so hopefully it'll work for everyone else as well.