Fraps is the best.
Look up ffmpeg to convert to FRAPS RAW to smaller files.
It will take a binary script, but it's worth researching!
or Camtasia if LoL can run in Windowed mode. Though, with Camtasia, you do NOT have the option of removing your mouse cursor in game.
If that doesn't bother you, then Camtasia is less resource/space intensive.