How do I delete old Time Machine backups on an external drive for a computer I no longer own?

The simplest thing would be to erase the drive and start a new set of backups.

I would more likely stash the TM Backup away and purchase a new drive for the new computer (assuming there IS a new computer). I'm also assuming that you have already migrated your data to the new computer (again, assuming there IS one). Typically, your data is FAR more valuable than the cost of a new ** drive. (If it is NOT, why bother backing it up to begin with? ;-)

You first have to remove the drive from Time Machine

System Settings > General > Time Machine. Select the drive and click the - sign.