I have just replaced my Samsung Galaxy Tab3 battery and charged it up. Works fine and all but powered it off and came back to it the next day and well it was dead. I had to hold the power button and down button to get it to do anything then plug it in to charge it. It will not stay charged for more than a couple of hours even when powered off.
Sounds like you got a bad battery. If it is an aftermarket battery that doesn't shock me. While aftermarket parts can be cheaper they also tend to be problematic. If that is the case, I would try returning it, get your money back and get an OEM battery.
Now if it is an OEM one, then I would have them exchange it for another. It just may be faulty. If you got it through another place (ebay, etc.) then it may well not have been new.