Repaired my sister's grandson's Tab 3 which had the same problem, won't show it's charging, won't power on. You have to remove the back cover of the tablet and disconnect the battery cable from the circuit board, then reconnect it back, and put it on a charger. The display will flicker/pulse when connected to a charger, and after a while will show it's charging because the battery was ran till dead. My advice if you're using a mobile phone charger, generic charger, or computer usb port is soon as you can power it on turn off the wifi and then turn off the screen (hold the power button till the screen goes dark) but if you have a 2.1 amp tablet charger you won't have to do either and can leave it as is since it can supply more power than the tablet is using (can't say that about the prior three). I posted a youtube video about this on another post.
If this happened to you, I may have a possible solution:
1) you let the battery die all the way
2) you are using a non-factory charger
I just went to Samsung website and the first thing they ask you to do is not try to charge with a non-factory charger.
Well, I went through my mountain of chargers and found the white travel charger that came with the tab 3, the moment I plugged it in, wammo, it started to show charging icon. Once I let it charge for a couple of minutes, which brought it to 5 percent charge, I switched to my multi port non-factory charger and it kept charging without any issues.
Lesson learned:
1) don't let your tab 3 drain all the way
2) if you do, you will need the original travel charger to bring it out of its charge coma.
Hope it helps...