Having hard time backing up with external disk on notebook

Launch Disk Utility. In the left window pane select the TimeMachine drive. Look over in the right side for the TimeMachine icon and you'll see something like USB External Physical Volume - Mac OS extended (or however it is formatted) You'll also see a bar chart indicating how much space the drive has and how much is in use.

How is the TimeMachine drive formatted? It should be HFS+ (MacOS Extended (journaled)) Is it?

If you've installed software for the drive from the manufacturer (e.g. Western Digital), remove it. It is not necessary and, as you've discovered, will cause problems.