How do I build an outdoor cat feeding enclosure to keep flies out?

Like a smaller catio?

Hang fly tape nearby to catch them.

i have seen chickenwire used to build a cat tunnel!

I'd recommend not keeping cat food outside, just because it will draw mice and other critters, even if you manage to keep the flies out. If you need to keep the food outside for some reason, I'd try something like a treadle feeder that's used for chickens. When the animals stands on a platform in front of the feeder, the lid opens. It's built so that smaller creatures can't open it. Not sure if you can train a cat to use one of these, but if you can get chickens to do it, I'd assume you could teach a cat.

You’ll have to use insect screen and staple it to your box frame. I’d make it out of treated 2x4’s unless you don’t expect it to last more than a months.

There are fly traps from your local Garden hardware store. They look like a plastic bottle and they put a bait inside of them. But I would keep it away from the food so that it’s not drawn to the food