I noticed some threads asking about this, and I looked ever so briefly for a thread discussing how to accomplish it, but I didn't see one, so I'm going to make a super quick tutorial for you guys so you can play fullscreen windowed mode like I use when I'm streaming. Check out what it looks like.
What you need is a little program called shiftwindow. This is a super easy to use program that automatically centers the window on your screen, so you can play fullscreen without having a border around your window.
THIS ONLY WORKS if you're playing at the full resolution of your monitor.
When you download and install shiftwindow, you get a program box that looks like this.
When you start a game in League of Legends, you need to make sure the game is set to windowed mode, and that it's at your monitors full resolution. If you do that, your game window will have the bar at the top, and by using shift window we can get rid of it.
What you'll do is AFTER you start your game, you bring up shift window, and you hit the "Grab" button by window title. That'll open a little menu, and you select the "League of Legends (TM) Client".
After that you hit the "trigger it" button and you should be all set!
Let me know if you have any questions!!