You were able to use iTunes to update it . . . did the Mac or PC also charge the phone? If so, maybe the problem lies with your charger. It could be defective . . . and some cheapo generic chargers don’t have the firmware that the iPhone needs to talk to.
I had this issue, and solved it by plugging into my iMac with iTunes running. Entered recovery mode by pressing the home and wake buttons for 10 seconds, and the phone stopped cycling and started charging. Actually doing an update or recovery is optional, but it’s not a bad idea.