iPhone not turning on when pressing lock button/erroneous charging When I try to turn on my phone by holding the lock button, it doesn't turn on. It just stays dead. 

But when I connect the Lightning cable, all it does is show the battery symbol, then go to the Apple logo, and repeats.

What should I do?
Let it charge overnight, if the behavior is the same you could try another charger and/or cable as it is possible that you have a bad charger or cable. 

Failing all of that it is likely that the battery in the phone is kaput and needs to be replaced.

If you are in warranty, take it to an Apple store. If it is not in warranty you could do the same, or if you know of a reputable phone repair shop they can do it as well.