Somebody may encounter black screen or blank screen after iPhone 5 LCD screen get replaced. Everything is fine, phone vibrates when connected to iTunes. There may be three possible reasons:

You broke the flex cable during the installation.
The metal bare connections around the flex cable can ’short-out’ the screen. Cover with Kapton tape to avoid any ‘short-out’.
Your replacement assembly does work but you have trouble on the backlight. Check if you still be able to see the logo with enough light by holding your phone under a lamp. If so, check the back light filters FL2025, FL2024 and FL2026 and replace it if you find any sign of damage. you can find them out around display connector near BS2503