I downloaded ComboFix and the icon is on my desktop. However, when I double click it it simply shows me a bar filling with green, and once the bar is full, nothing happens.

The program doesn't seem to be doing anything at all. Anyone know why it's stalling like this?

Unfortunately the author of the tool does not want information on how Combofix works on public forums.
Please read Combofix's Disclaimer. Using this tool incorrectly could lead to disastrous problems with your operating system such as preventing it from ever starting again.

The only public information that is available can be found at this guide:

How to use ComboFix