Put enough water in the tank to compensate for the hot water tank filling as well (if the valve is open).
Open a cold faucet.. or all of them.. and turn the pump on. You should hear air escaping or water spitting out.
Check the 'city water connection' outside and see if its leaking air or water. This is common, and if it does the check valve there is bad.
The pump turning off by itself for a time period makes me think its over heating, which prob means water is not being pumped thru it. Check to make sure you didnt close a valve that should be open. Follow the pump line back to the tank as far as you can.