The camera should allow the recording at high resolution as well as high number of frames per second. It should also allow the transfer of feed to the computer at high speed.
Why a camera and not an eyetracker? Obtaining images of eyes is just the start of the process. There's the issue of stimuli and synchronisation between the stimuli and the recording system; calibration; detection of blinks and other artefacts. Particularly if you want millisecond time resolution (necessary if you want to do analysis of saccades), you could obtain a camera capable of 1000fps (for a price), but end up with no workable system. If all you want if fixation tracking (as distinct from eye tracking) then a relatively cheap, low temporal resolution eye tracker would be a good solution. It depends on exactly what types of eye movement behaviour you are interested in, and in what circumstances.