how to increase no. of led in led cheaser circuit.

A schematic diagram of the circuit would be more helpful than a photo of the pcb.
However: how are you going to connect the LEDs?
In series will require 1000*1.6V=1.6kV (assuming a red LED with 1.6V voltage drop). This voltage can't be handled by the circuit, not even by changing a transistor.
In parallel will require 1000*5mA = 5A (assuming each LEd is operated at 5mA current). A power transistor e.g. a TIP3055 with a suitable heatsink can handle this current- You will need one current limiting resistor per LED.

A viable alternative is a combination of series an parallel circuit: E.g. put 10 LEDs in series, add 1 series resistor for current limiting. Each string of LEDs will have a voltage drop of ~16V (assumption as above). The current (ILed) can be set to e.g. 5mA or 10mA as per your requirements and the LED's characteristics. Add the voltage drop across the current limiting resistor to get the required operating volateg (e.g. a voltage drop of 2 V will require 18V power supply).
Put 100 of these series circuits in parallel. The transistor needs to be able to deliver 100*ILed where ILed is set as discussed above. Assumin 5mA, you now have 500mA.
Combined you have the requirements 18V/500mA. A 2N2222 can handle this.

 

An LED doesn't drop current. An LED is controlled by current. You have to set the current externally (e.g. resistor ot current source). Read the sticky post on LEDs in this forum.

Assuming that your LEDs drop 3V at 20mA, you cannot connect 4 of them in series, becaus that means you have no spare voltage for the current limiting resistor. If you put only three of them in series, you're wasting a lot of power in the current limiting resistors. What type are the LEDs? Do you have a link to the datasheet?

Assuming we find a way to operat 4 LEDs in series at 20mA, this means that 1000 LEDs are made from 250 strings of 4 LEDs each. So the total currrent is 250*20mA = 5A. The 2N2222 is not suitable for this. The TIP3055 is usable. You should use a heatsink to get rid of the thermal power. YOu can use any other NPN transistor rated for Ice>=5A and Vce>=15V

 

How would you use a diode to increase the number of LEDs?

The photo is insuffcicient. Sorry I didn't express myself more clearly: We need a datasheet of the type number. The LEDs on your photo could be any type and have any characteristic.