It depends. Drivers, the encoder you use, the video card you choose, all these will affect whether and how much the video card will be involved in the encoding.
With cell-phone video, the quality of the stream is likely low enough that unless there is a particular effect you need, it will not be economically feasible to purchase a new video just for this purpose.
Unless you're working on a professional project, just use Handbrake and the hardware you currently have. Install the latest drivers for your mainboard (chipset), video card and sound card, to make sure they can all function in any way the software will utilize them, and just use what you've got.