As you can see by reading stimmer's code, he declares 4 consecutive buffers : uint16_t buf[4][256]; then the PDC DMA (Direct Memory Address) fills 256 samples in the first one buf[0], and before the end of the first buffer, the address of the second one is passed to the PDC DMA into ADC_Handler() to prepare for the filling of the second one, and so on....The process of a PDC DMA is not obvious at first sight.
Now if you want to keep the 256 samplings of buffer[0] before it is overriden, you can store them elswhere into the SRAM, or an EEPROM or the Flash (DUEFlashStorage lib) depending on the amount of data you want to keep.
Do that in loop(), instead of a SerialUSB.Write().
As already stated, the best place to learn is the SAM3X datasheet. Even though the SAM3X datasheet is ~1400 pages, reading through it is the best way to learn how to use it. It's intimidating, but as you go from chapter to chapter you will see that there are a lot of similarities in the way peripherals are setup and used.