The problem is your type of connection. The coax cable is an RF type connection. The DTV box is sending out an analog TV signal on channel 3 which you connect to on the small TV.
A computer monitor will not have a TV tuner built in, so it won't be able to use that connection. You have a couple of options.
Option #1: you could buy a 20-26" TV instead and use the tuner.
Option #2: If your DTV box has an HDMI connector, you could buy a modern computer monitor and connect via HDMI. The problem is that most monitors don't have speakers or volume controls for audio, so you would need to set up an amplifier and speakers in addition. That is probably more trouble than its worth.
Option #3: You could buy RF to RCA adapters for the coax and connect to the composite (yellow) output of the DTV box to the Yellow input of a monitor. Not all monitors have component inputs though, and you still have the audio problem.
Option #4: Upgrade everything to a HD sat box, a bigger monitor and connect via HDMI.
Of the 4 options, the simplest one is probably to simply get a bigger flat panel TV in 19-26" and use the existing coax.