What is an effective way to attach this plastic filter to aluminum without drilling?
I would suggest good old J-B Weld for joining plexiglass and metal, and thistothat agrees.

Rough up the mating surfaces with 120 grit sandpaper and be sure to clamp where you can. If you don't want the J-B Weld to show, you'll have to put the plexiglass angle above the aluminum channel. However, it will definitely be harder to get a good bond, you'll have trouble clamping it from what I can see of your design. If you don't care because it's in a closet, you could cut some scrap wood to length and place it underneath as a brace while the J-B Weld cures.