Stove hood filters are one of those things that people forget about cleaning regularly because they don't usually see them. But they can get pretty nasty, and even dangerous, if you don't take the time to clean them.
If you remember to do it regularly, you can just stick your filter in your dishwasher and run it with hot water. That's assuming it fits in your dishwasher. But if you've negelected yours for a while, it's probably a greasier, grimier mess than your dishwasher can handle. One Good Thing by Jillee suggests grabbing your deepest pot, boiling water with about a 1/2 cup of baking soda, and plopping in your filter for ten minutes or so. Most filters are long, so you'll probably have to do half a side at a time.
How To Clean A Greasy Stove Hood Filter [One Good Thing by Jillee]