I had a similar problem just recently. Do the replacement (ifixit parts + tools). Reassemble, slide the back closed, screws. Plug into the charger. You see the apple, then it goes black. Then apple, then goes black (bad). Put the old battery back in, plug in, no problem. When you plug in the charger you see the empty battery and it’s charging (good). My problem ended up being that pressure switch. As soon as that contacted the back of the phone, it would go dead. So new battery in, don’t replace the back of the phone, plug in the charger, phone boots up, 50% charge in the battery. As soon as you put the phone back on and slide it closed, phone goes dead. Plugging in charger gets you the apple-black-apple-black again (phone won’t boot). Pop the phone back off, plug in, phone boots fine.
Note the pressure switch worked fine with the old battery and clip. Not sure what’s up with the new one. It might be fine to just leave the pressure switch out or make it so the pressure switch can’t contact the metal part of the phone back which is what I ended up doing with a small piece of the sticky clear battery extraction material (you pulled on the clear tab to get the battery out).
Could be the orientation of the pressure switch gold contact. It looks pretty square in the picture but maybe needs to be as far toward the battery and as far to the top of the phone as possible (the range of works/doesn’t may be small).
Before blaming the battery, try it without the phone back installed and see if the phone boots up ok. Try not to get zapped.
Note: I’m not using this 4s as a phone, just a wifi enabled IPOD with a screen and a web browser.