The tuner in your new TV card is basically the same as the tuner integrated into your television set - both require a cable box to convert the new digital signals back into the old analog format and output the result to channel 3/4.
Reconnect your cable box and plug its output - preferrably S-Video or composite if possible for superior quality, but standard coax will work fine, too - into your new TV tuner card. Now when you tune the TV card to channel 3/4, you should see video.
The only other problem is that your computer has no way to change the channel that the cable box sends to the TV card. There are three separate ways to work around this:
1. Just use the cable box's remote to change channels manually. This makes scheduled recordings a hassle because you have to remember to leave the box tuned to the proper channel before the recording starts.
2. Get a serial cable to connect your cable box to your computer. You'll have to find the cable box model from a sticker on the box and do a search on that model number to see whether it supports a serial cable, which you can purchase online.
3. Get a USB-UIRT which your computer can use to send IR remote signals to the cable box. Basically, you point your cable box remote at the USB-UIRT and press the buttons as directed so that the USB-UIRT "learns" the signals. Then you point the USB-UIRT at the cable box and leave it there; whenever the computer needs to change channels, it simply tells the USB-UIRT to send out the previously learned IR code.
(For more specific help/advice, it's probably best to post on a HTPC forum.)