Since you're playing LoL and presumably want to capture both the lobby and the game, the best way to do this is to create two scenes, one with a game capture source for the game and the other with a window capture source for the launcher/lobby. Monitor capture performs super bad in Windows 7 so you should avoid it completely. Aero needs to be enabled for window capture to perform well. Once you have your two scenes set up, you can use the Simple Scene Switcher plugin to automate switching between the two scenes -
https://obsproject.com/forum/resources/simple-scene-switcher.29/
Under Settings > Video you should set the base resolution to 1680x1050 and then the resolution downscale to 1120x700. Your monitor doesn't run at a 16:9 resolution so you won't be using 1280x720.
When you add a game capture source make sure LoL is running (the game itself, not just the launcher) and be sure to point the source at the correct process. You could also assign a game capture hotkey instead and press that once in the game.
If you continue to get a black screen for game capture, make sure you don't have any other programs running that could also be trying to hook into the game. This includes any programs that can show some type of overlay in the game - EVGA Precision, MSI Afterburner, Razer GameBooster and Synapse, Fraps, Dxtory, TS3/Mumble Overlay, etc. You may also want to try adding OBS to your antivirus software's whitelist, as some antivirus programs like Norton and AVG have been known to interfere with OBS.
