Phone boots to apple logo but turns black, not recognized by computer

You've got a lot going on with your poor old 5C, don't you?

It's hard to give you an exact diagnosis due to the fact that it won't proceed past the Apple icon on boot, so we can't get the phone to help us out with any clues.

I've seen this sort of behavior on an iPhone X, and the fix turned out to be to replace the power button/flash/microphone flex cable on that phone. So it could definitely be something like that on yours, where replacing a particular part would get it going again. My suggestion would be to start out with the iFixit troubleshooting guide here.

iPhone Keeps Restarting - iFixit Troubleshooting

You've done a fair amount of the things the suggest at the beginning of the guide already, but you do want to double-check your power supply, either by using with another phone or trying a different one altogether.

After that, the guide goes on to suggest the two things that were the first ones to come to mind to me, which would be to replace the battery and the lightning port flex cable assembly. The nice thing about an older phone like yours is that you can now get the parts for it dirt cheap, so if it was mine I'd just order both parts and replace them at the same time.

At that point you can reevaluate whether you've made any progress or not, and while you have the phone open you can also try unplugging the flash/button connector from the motherboard to diagnose whether that's what's keeping it from booting - that's the way I pinned down the problem on the X I was working on. With the cable disconnected, plug in the power cord and see if there's any difference in behavior; if so, you've probably got the cause pinned down.

Of course, once again since the parts are inexpensive, you might also just go ahead and order that assembly along with the other parts if you wanted to.

Another alternative would be to swap out the motherboard into a different functional phone; that would give you a very quick go/no go diagnosis as to whether the problem was on the motherboard or not. I would imagine you could probably pick up an iCloud locked 5C on eBay or maybe OfferUp pretty cheaply these days.

Anyway, hope that's given you some food for thought; be sure and let us know how it all turns out so the rest of us can benefit from whatever you learn about this problem.