As far as I know, installing on a headless computer is something which can be done using a variety of tricks. The first of the following suggestions uses accessibility features to compensate for the lack of screen. The second one is the easy way to pretend you know the exact sequence of keys to press. The third and fourth are one of many possibilities for installing over a remote connection.
Debian has accessibility features directly in the installer, and it beeps when the installer is ready. After that beep, you simply have to press s and Enter to get speech synthesis, which you can use if your old machine has an audio jack. See https://wiki.debian.org/accessibility#Debian_installer_accessibility for more details.
Install in a virtual machine at the same time as on your old machine, and be sure to press the exact same keys.
Install via SSH, as noted by @FaheemMitha in the comments. If you're willing to try out something new, NixOs is pretty easy to install over SSH from an existing system. It then is just a matter of booting off a Live USB where you have already configured ssh (add your public key to the ~/.ssh/authorized_keys, with chmod 700 for the folder and 600 for the file), and log in from another computer. Don't forget to set up SSH in the freshly installed system, before rebooting :) .
- If you have a (couple of) USB to serial dongles, you can always try to pretend you found a time machine, and run the installer over serial (Debian supports that, I think, as well as a lot of other distributions).
Finally, you can install the operating system on another computer, as Fiximan suggested. An easy way to do this without removing the hard disk drive is to install in a virtual machine (QEMU comes to mind) using a raw disk image, and then simply
dd that disk image onto the harddisk (which can easily be done by blindly typing the command on the headless computer from a live USB: press Ctrl+Alt+F1, log-in (check the username and password beforehand), and type
zcat /path/to/img.gz | dd of=/dev/sda. And hope that
/dev/sda hasn't been mapped to your harddisk and not to the USB key :) .