I had this same problem after replacing the battery: all the inner parts were connected with a ‘snap’, screws were in right places, screen secured in the casing… but the screen was black. I felt the iPhone 5s heat up when I pressed the power button and home button at the same time, and the heat was still there when I plugged the charger in. Sometimes I could see a very dim purple-ish backlight when I pushed both buttons together, but the screen stayed black. I spent the whole day agonising about this, because the iPhone was working, it could turn on, it could heat up, it could vibrate, but the display couldn’t show anything. I was this close to giving up.
Stumped, I popped the screen off yet again to check the internal connections, but this time around I didn’t do the last step: pushing the screen back into place. I pressed the home and power buttons, and the white Apple logo showed up! Phone works, display also works!
I moved to push the screen into place, but then I saw the display fading out into black. Lifting the screen slightly returned it to its unblemished state. Pressing the screen down again only caused it to fade to black again. I checked and tried pressing each corner of the screen down to see what section of the iPhone was causing the blacking-out… Pressing the top-left corner seemed to be the cause. The components on the underside of the screen that corresponds to the top-left corner was pressing onto the top edge of the replacement battery.
The replacement battery is slightly slimmer in width and height to the original old battery I had just replaced. Because of this millimeter-difference, the replacement battery has a bit of extra room in its compartment. The first time I positioned it, I had it placed closer to the top, but this positioning caused this very black-screen problem. To fix this, I had to reposition the battery and scoot it closer to the bottom of its compartment. This will open up that millimeter of space the top-left corner underside-screen components needs to avoid jamming itself into the top edge of the new battery.
From there, it was just a matter of pressing the screen closed and turning the phone on, no black screen anymore!
Hope this helps anyone who needs it.