Making a USB port transmit wirelessly via bluetooth
Consider I connect a 5v battery to the Vcc of the module and somehow
  connect the HC-06 module to the usb of board. Will it be enough to
  power the board to work?


Yes, but only if the battery puts out exactly 5V. No battery chemistry is capable of this, so any 'battery' that does it must have a built-in regulator to stabilize the voltage. 


  Somehow connect the usb of board to the RX of the module. Or connect
  the usb of the board to ardunio(if that makes sense I don't know how
  to do this). Then connect the usb module to the ardunio.


You cannot connect the Powerboard interactive display directly to the HC-06 because they use vastly different communication protocols. To talk to the Powerboard you need a USB host that emulates a Windows PC running the Powerboard USB driver. Theoretically an Arduino could do it, but without knowing the exact protocol used it will be impossible in practice. 

The Powerboard has a custom USB driver that only works in Windows. However it appears to be based on a standard Silicon Labs CP210x USB-serial interface, so it might be possible to remove the USB interface from inside the Powerboard and access the serial data directly. This data could then be sent via an rf link to the PC, where you would feed it into the USB interface that you removed from the Powerboard. This is advanced hacking with no guarantee that it will work...