Esp8266 connection and testing

Am trying to connecting ESP8266 to arduino uno and testing AT command but its not working.Tried everying even to use external power supply.Any help please.

In order to get some good help and suggestion is to provide code examples of what you have tried and how you are attempting them.

Any pics of how you have things wired to the board and maybe the sketch you are trying to use would help get you the information and help you need.

am doing humidity and temperature logger via thingspeaking by using Lcd,esp8266,DHT11.

circuit connections of esp8266

i connect the module's VCC pin to the positive rail of your 3.3V power supply, GND to the supply's ground rail, TX to LV1 of level converter, RX to LV2, and connect CH_PD to one end of a 10k resistor, the other of which is connected to +3.3V rail.
The LV pin of the level converter is connected to +3.3V, GND to supply ground.

On the other side of the level converter, I connected HV to Arduino's 5V output, GND to an Arduino GND output, HV1 to RX and HV2 to TX.

After that i upload empty program but nothing was running on serial monitor.when i tried to write AT command and press send,AT command was disappearing.

from what I can tell you hooked it up correctly so it should actually work.
here is a good guide that helped me a long time ago when I was stuck at the same point:
http://rancidbacon.com/files/kiwicon8/ESP8266_WiFi_Module_Quick_Start_Guide_v_1.0.4.pdf

I would actually recommend not using the ESP8266 with AT commands. It is (or used to be) unstable and very frustrating to use. Also you should know that the ESP8266 has a lot more computing power than the Arduino but has less IOs. The ESP8266 can also be directly programmed using the Arduino IDE. It is a lot more complicated to get it up and running but Ivan Grokhotkov is doing a great job!
Read all about it here:

To get it running more easily you should get a NodeMCU board. They are cheap and have a (mostly) stable power supply built in.

i tried the connections AT command is working eccept AT+CWMODE command.Any idea on how to fix it?