iPhone 5s not turning on

I have an iPhone 5s. I replaced the cracked screen and the battery very recently. A week ago I noticed that my camera and flash weren't working, so I got my partner to pull the phone apart again and found that the camera wasn't plugged in to the board. So we clicked it in and turned the phone back on (with no worries). We checked the camera and flash and the problem was fixed but then it ran out of battery. Plugged it in and it was charging. Left it over night and woke up to it still on the charging screen. I click the top button to turn it on and the Apple screen comes on. It stays on the apple screen for a little while, flicks to a dark grey screen for about 2 seconds and goes straight back to the apple screen... Then it vibrates twice as if its turned on but then dies again and goes back to the charging screen... Sometimes it'll do it twice, maybe 3 times without me even touching the phone after that first time, and sometimes it'll only do it once...

Would this be a:

- Battery problem?

- Charger problem?

- Wiring problem?

Need to keep phone as it's got all my photos, vids and other important stuff on it. Any help would be greatly appreciated!

Kind regards,

Criddle