Leviton Dimmer Switch Model DW1KD
From the description provided for the existing old switch it sounds like that was wired as a simple switch loop. As such there is no NEUTRAL in the switch box. The HOT black lead is bringing the line into the box and the other black wire is connecting out to the load (light). That load black wire may actually be a white wire that has been sleeved with a black tubing, painted black or taped with black electrical tape. 

From the installation sheet for the DW1KD the switch is a smart switch and will require a NEUTRAL in order for it to function. 



You will not be able to get this to work with the existing wiring. Since the HOT is clearly off in the electrical box at the light you will have to work out running a new cable from the light box to the switch box that has a four conductors (HOT - Black; NEUTRAL - White; LOAD - Red; GND - bare wire).

From the above diagram note that the Red wire of the new switch goes to the load. The Black wire on the switch goes to the HOT entering your switch box. The White wire from your new switch has to be connected to the new NEUTRAL that will be provided by the new cable that you need to run.