You will need one of two things for your computer: Either a new video card that has a VGA and/or HDMI port in addition to a DVI port for your current monitor (~$40 for a cheap one) or you will need to get an adapter.
There are various kinds of adapters that allow for DVI to HDMI, DVI to VGA or DVI to 2 different types. If you have 2 DVI ports on your video card for 2 monitors, get an adapter for one and then the corresponding VGA or HDMI cable to plug in the TV.
If you only have a single port on your current video card, then you will need a splitter adapter that goes DVI to DVI+VGA or DVI to DVI+HDMI. You plug in to the port you're using for your current monitor, plug in that monitor into the DVI side of the splitter and then plug in a VGA or HDMI cable from your TV input to the VGA or HDMI side of the splitter, depending on which you get.
So to review, you need either a new video card that has multiple output ports or a splitter adapter + VGA or HDMI cable based on which adapter you get. You will need a VGA or HDMI cable for the TV regardless of your choice, but getting a cable and adapter/splitter is the cheapest option.