If your DVD player has a Component output (Red/Green/Blue + Red/White for audio), that would be the way to go. Yes, you'd need to purchase a cable. If it only has a composite output (Yellow + Red/White for audio), then that is the way to go. Since you already have that cable, I'd just go with that, rather than spending money needlessly. Connect whichever DVD player output you have directly to your HDTV. Your dish box does not have any inputs so running it through is impossible.
When you want to watch a DVD, you will need to change the input source on your HDTV from HDMI (again, assuming this is how your Dish box is connected) to whichever input source your DVD player is connected. Here are the input sources I have on my Olevia 232T 32" HDTV:
HDMI
VGA
VGA-Component
Tuner
Composite (<-- if you connect your DVD player to the TV using the Yellow/Red/White cable, this is the input source you want).
S-Video
Component
-Wolf sends