You haven't mentioned the version of Android running in your phone and I assume it is KitKat.
If so, you can use ADB (Android Debug Bridge) tool to record the video. The are many blog sites that explain how to do this and one such that came out first on my Google search is this one. Even Android SDK page on ADB tool commands explain this well.
In gist, you should use these commands:
adb shell screenrecord /sdcard/demo.mp4 adb pull /sdcard/demo.mp4
The first one starts capturing the screen to the specified file and the second one pulls (copies) the recorded file to your computer.
Update: By the way if you are finding it difficult to download the entire SDK just this small ADB tool, you can use the one from this XDA page.