I presume you have a Windows PC with a decent internet connection.
1. Download the DSL iso image from one of the download mirrors listed on http://damnsmalllinux.org/download.html
(you'll need to browse to the subdirectory 'current' and select the file 'dsl-4.2.5.iso') and save it.
2. Burn the iso image to a writeable CD using whatever CD burning software you've got on your Windows box. I have Roxio Creator Classic on mine, in which all you need to do is select 'File > Record disc from image', browse to the iso file and click OK. If you get stuck, use the application's Help facility and search for "iso", "image" or similar, hopefully there will be instructions in there somewhere.
3. Power up your old PC and get into the BIOS (this is usually done by hitting Del or F8 while it's booting), and make sure that the boot order settings specify that the CD-ROM should be used as the first boot device, with the hard disk as the second.
4. Pop the DSL disc into the drive and reboot the PC. I don't know anything about DSL, but normally an installation environment will boot from the disc, and a series of prompts will guide you through the installation.
At some point, the installer will probably ask you how you want to partition and format your hard disk. I presume from your initial post that if there already is an OS on the old disk, you don't want to keep it, so just select the option which utilises and formats the entire disk.
You will normally have the option to configure your network card during installation, assuming the installer can find it. Provided the card is a standard PCI card, DSL should find it easily, and then you can either configure it to use DHCP (if you've got a DHCP-serving device on your network) or with a static IP configuration.
The installer will ask you to specify a root password (sort of equivalent to the Windows Administrator password) and at least one 'normal' username and password.
If the installation program asks you any other questions and the answer isn't obvious, just accept the defaults.
That should do the job - DSL should install, and when it's finished you just pop the CD out, then log in with the 'normal' username and password that you entered during installation, and explore the wonderful world that is Linux
Please note that, as I said earlier, I have never used DSL, but I've installed and messed about with dozens of Linux distros and they all usually install in more or less the same way.
Good luck - let us know how it goes!