For a no power problem you're right to think battery, and then new dock connector.
A new battery should turn the phone on. If not, then you may have a different problem.
First rule out bad (new) battery, then dock connector. After that we will start thinking about the board. Make sure to test if the phone is on by connecting it to iTunes---that way a no display issue won't get in the way of your troubleshooting
jessa