Power supply issues with LCD and MPC23017 expander chips
Use an external 5V regulator to increase the amount of current you can use. The built in linear regulator for the Arduino is limited based on the input to output voltage difference causing high power loss and heat. Keep in mind that the Arduino also uses about 50 mA or more depending on your code.

The other option is scanning, that is turning the leds on and off rapidly so that it looks like they are all on but only one is really on at any given time. This would reduce your average current draw.