iPhone 5 Stuck In Bootloop

I would recommend starting with a new battery, if you buy from a reputable shop you should be able to return it if the battery wasn’t the problem.

Bootlooping can be caused by more things as well, have a look at thins answer: Stuck in boot loop

There are a number of things that can be wrong, starting with software corruption AND/or hardware failure.

What everybody here is saying: you must first try to restore from DFU.

1/ Disconnect the front camera, leave it off.

2/ Connect a known good battery. Known good, not necessarily new.

3/ Connect screen (LCD + touch connectors only).

4/ Connect home button.

5/ Connect to computer. Turn on iTunes.

- Restore from DFU:

6/ Press and hold Power Button for 2 seconds.

7/ Keep holding Power Button and now press and hold Home Button.

8/ Keep holding both buttons for 15 seconds.

9/ Release Power Button only.

10/ Keep pressing Home Button for another 12-15 seconds.

Do you see a black screen on the iPhone and on iTunes the message "iPhone in restore mode detected?" If yes, click on restore.

If not, report back for other steps.