How do I show my Android device screen on my PC with no lag (at 30fps) Device: Galaxy Note 3 N900W8

I need a solution that would let me show my phones screen on my PC at a high framerate. I tried VNC. It could not find a display method. I tried some random apps that work with USB debugging. No dice. Any suggestions. HDMI is not a option because then I would need to purchase a capture card and a HDMI adapter for my phone.
You mentioned that video capture is your goal but does that mean that it has to be live on your PC? If what you're looking to do is get a capture of your phone's screen so that you can post it to YouTube or something, consider trying to capture locally on the device then transferring that video file to your computer for processing and upload. This might give you better results than trying to "cast" your phone's screen to your PC live and then capture it there.

If your Note 3 has been upgraded to Android 4.4.x Kitkat you can use adb's built-in screenrecord utility. Full documentation is available from Google Android Developer Support. Here are the basics:


  To use the screenrecord from the command line, type the following:
  
  adb shell screenrecord /sdcard/demo.mp4 
  
  Stop the screen recording by
  pressing Ctrl-C, otherwise the recording stops automatically at three
  minutes or the time limit set by --time-limit.


Google also outlines the following limitations:


  
  Some devices may not be able to record at their native display    resolution. If you encounter problems with screen recording, try
  using a lower screen resolution.
  Rotation of the screen during    recording is not supported. If the screen does rotate during    recording, some of the screen is cut off
  in the recording.
  Audio is not recorded with the video file.