T6963C with Arduino Mega 2560 Interfacing Issue.

As you said i adjusted trimpot wiper but screen back led brightness is not effecting.

This is not related to the LED brightness, but instead the contrast of the LCD itself. In fact you should see the transparent/black change without background LED active, ambient light should do the job also.

Should i increase pot value from 10k to 20k or 50k?

10k is fine, 20k should be also ok, 50k might be a little bit too high.

Take care of VEE which generate -ve voltage (Around -15 v to -25v) am getting -23.8v. Means Display is Healthy.

It just means, that the voltage converter is ok.

AT LEDA pin am getting 4.9V even if i not connected anywhere. So should i connect to 5v or leave the connection as free.?

You should be able to see something on the LCD without backlight, so I suggest to postpone the backloght LED problem.

Oliver