Hmmm... Well, you can certainly try. HTPCs have been around for a long time... even before the circa 2005 machine you are wanting to use. The only difference is the performance expectations are higher now.
Are you connecting this computer to a TV? If your TV doesn't have VGA as an input, you'll need a cheapo video card. I don't know... $30?
I'm assuming you are wanting to record cable TV, right? That will require a cable card tuner. You have basically two options... either the USB version of the Ceton InfiniTV4 tuner, or the Silicon Dust HDHomerun Prime. You would not be able to use the Internal/PCIe version of the InfiniTV4... you only have one PCIe slot and that will be taken up by a video card. The cable Card tuners themselves will take up around 3/4 of your budget. If you get a USB tuner, you will need to find a PCI card that will give you USB 2.0 ports. I don't believe your system has USB 2.0. That's probably another $10-20.
Now, to record cable card you will need Windows 7, or a special version of Windows Vista. That's a few bucks more... Upgrade version from XP (I'm assuming your Compaq has XP on it now) might be $50?
In order to install Windows 7, you'll need a minimum of 1GB of RAM installed. Once it's installed, it will run on just 512MB. How much does your Compaq have?
I commend you for wanting to tinker and learn, but some of the money required to get this thing going is a waste. You can use the $200 and get a cheapo Sempron/Llano/Trinity/Celeron/Pentium based system for that. You can reuse the case (maybe... Compaqs used to be weird and proprietary) and hard drive from the Compaq, but you could easily get a CPU/Mobo/RAM for $200.
Check out some of the threads here. People ask about cheap builds every single day.