Activating Relay Coil (NPN and MCU)
I had same problem too.

First assume your coil needs 100ma to click. Then your hfe is 325 which is the amplification factor in common base mode. That is the current from your base will be 325 times smaller than the current from your collector.
You told you gave 10k resistor in between them. Assuming that esp is running exactly at 3.3v and the voltage drop across the base to emitter is 0.7v we get a potential drop of 2.6v across the resistor. This led to a base current of 2.6/10000 = 0.25mA.
This will result in a collector current of 0.25*325 = 81mA. Which is barely enough for the required specs. 

You can either use s a transistor with higher hfe or lower the resistor down to 6.4k or 5.6k then it will work.