Without knowing what connections are available on the cable box, no further discussion is possible. Do they have a web site where you can get a manual download or any pictures? I have no doubt that in order to get the same functionality you had with satellite, you will wind up paying more.
The cable box must have either a s-video or a composite output, or even component. If it only has ONE output, then you have to split or switch that output to the monitor and to the recorder. The simplest approach for splitting or switching the video output is usually with a stereo (home theater) receiver where you can connect the recorder like you would any recorder, with an input set of connects and an output set of connects. Then the recorder gets whatever the receiver is switched to monitor. The nice thing about such a receiver is that it also switches the audio, which must also be considered.
If the cable box has 2 outputs, then you use one for the monitor and one for the recorder.
You certainly don't have to use the cable company's DVR, but don't expect them to help you use your own.