Take a look at this YouTube help page which explains how to export your movie from Final Cut Pro for YouTube high quality.
Final Cut Pro: File -> Export -> Using Quicktime Conversion -> Options
In the Options box:
In the Video section select the following: Compression Type: H264 Frame Rate: 30 Data Rate: Automatic (if you choose to restrict the Data Rate, make sure to choose the highest value acceptable to you- more bits = better quality) Key Frames: Automatic Frame Reording: Unchecked
In the Video section click “Size”: Dimensions: Set this to the original size of your video (Note: HD resolutions preferred- 1280×720 or 1920×1080)
In the “Prepare for internet Streaming” section: Select the “Prepare for internet Streaming” and choose “Fast Start” from the drop-down menu.
Once you’ve made all of these selections, click the “OK” button to save your preferences.
Add a title to the video file and click the “Save” button to save your edited version of your movie. Your project will then be exported and be ready to upload to YouTube.
Note: The tips in blue are important. Please make sure you format your files accordingly.