Getting more current through an LED array
The current a bipolar transistor configured as a common emitter amplifier will allow from its collector to its emitter is equal to its beta time the base to emitter current. 

A beta of 100 is common for many transistors. Assuming this information is close enough to your transistor, we can choose a base series resistance suitable for the collector to emitter current desired.

For example if you were using a 5 volt microprocessor and the voltage drop across the base to emitter was the usual 0.7 volts for silicon semiconductors and you had a base series resistor of 10Kohms:

V = I * R
V = 5 - 0.7 = 4.3 volts
R = 10000 ohms
I = 4.3/10000 = 0.43mA
I(ce) = Beta * 0.43mA
I(ce) = 100 * 0.43mA = 43mA


You could pass up to 43mA from the collector to the emitter.  If that is not enough,you can lowered the base resistor to 1Kohms:

R = 1000 ohms
I = 4.3/1000 = 4.3mA
I(ce) = Beta * 4.3mA
I(ce) = 100 * 4.3mA = 430mA


And increase the current to 430mA.  You still need to verify that the processor can supply the current to control the transistor (4.3mA as calculated above) as well as pick a transistor of sufficient capacity to handle the load.

If a beta of about 100 is not enough, consider using a Darlington transistor arrangement. This type of "2 transistor" device has a gain of the product of each individual transistor's beta.  So, it is common for such a device to have a beta of 10000!  Using this device, you can use a much larger base resistor and still be able to switch more current.