I have shared my circuit diagram in the image i have attached in this post. The problem i am facing here is , the esp8266-12e does not start in first attempt, I have to reset the module (or power on/off) multiple times and then it starts after number of attempts. The voltage reading in multi-meter shows appropriate all the times, even the voltage in ESP8266-12e module shows perfect. but the module does not start.
The almost complete lack of decoupling in your schematic could be a factor. 100nF on the ESP supply rail is insufficient, during startup and transmission, the ESP can draw large current transients. I would start with 47uF and possibly more (and keep the 100nF). Additionally, you need decoupling on the PCF, at least 100n but 10u + 100n would be better, perhaps more depending on the size of LEDs you are driving.
Also, you haven't provided a picture of your circuit; that may give us insight into potential problems as well.
It is possible that some pins which the ESP checks for startup configuration are floating. I notice that you have GPIO2 tied to VCC; IIRC it is typically pulled up, shouldn't matter but may make a difference. Also, you have omitted the typical 100nF RST capacitor, that is also something you should add.
Finally, you don't provide details on your 3.3V supply. Is it possible that it is current-limiting during startup? If you have access to a scope, you should watch the 3V3 rail during startup to make sure it's not dropping out.
What are you powering the ESP8266 from? This is an important question since I had same problems when I had a very bad 3.3V supply. Use a 1117 fixed on 3.3V or a similar regulator powered from 5V to make sure lack of power is not the problem you'll encounter. Oh, and caps on the power line! (though ESP8266-12 has some integrated IIRC)
You have not mentioned which regulator you are using to provide the 3.3v. That may be useful to know.
In addition to the decoupling caps as others have mentioned, if you are prototyping this circuit on a breadboard you may check that the breadboard is not the source of your problems. On the ESP8266 forums many people have had a hard time delivering a stable enough power supply through low quality breadboards.