Ken:
The cable/TV/Windows industry have pretty much sewn up the copying of protected materials unless you can find a way to access "the analog hole."
Frankly, if you need to do it this week, use/borrow a videocamera and record it off the TV. A lot of cellphones will do this, although usually only a couple of minutes. Test it out first.
If this is a show of your dog on a local TV station, contact them directly about how to get a copy of the show. Most studio engineers will do that as a courtesy to their subjects.
If the station puts features on their website, you can often download the episode by hacking around on their website. or wait for someone else to put it on YouTube. "How to download video off YouTube" has a few million suggestions on the internet.
Years ago, I used a couple of different video adaptors and a Linux box running MythTV, but it was weeks of experimentation and constant tweaks to get it working. The good news is you can get a video in the format you'd prefer. But you earn it. 48 hours is a tad optimistic.