Can I use a separate power supply to charge the phone while working with a IOIO board?
No, it will not work this way. 

And the problem is not in common or "uncommon" ground - when you connect grounds of your power supply and your IOIO board, the ground becomes COMMON.

The problem is that your phone will be unable to recognize the cable as charging cable (because your IOIO does not provide any known charger signature on D+/D- lines), and the phone will limit itself to 500mA, which apparently is not enough, as you found, to run the phone in full-blown mode and charge its internal battery simultaneously.

To make the phone to charge and work, your IOIO board must provide some "charging signature" on D+/D- lines before is connects as a host to your phone, a signature that your phone will recognize. Normally it is done on board hardware level by designing in some port controller chip in-line with D+ and D- signal lines, under sideband control by the host microprocessor. Or the port can internally support USB Battery Charging specification (BC1.2), or QCOMM Quick Charge. Your IOIO board does not seem to support this functionality. 

If your IOIO board supports USB hubs, then you might be able to use one of products based on Microchip hub controllers, some of which do support BC1.2 on one or several downstream ports (charging ports).