To get the best quality out of Facebook, check its own video setting first.
Settings: In Facebook, go to Settings & privacy > Settings.
Media: Under Preferences, select Media, then Videos and Photos.
Video Quality: Make sure Video Quality is set to Optimized.
When you initially upload a video, it takes time for Facebook to post the high-res version of your video (especially a large one), because it processes in the background. Facebook will give you a lower resolution version to view in the meantime, making it seem like your video has not uploaded properly.
With a larger video, you sometimes need to tap the small HD mark on the video to see it in HD.