Is it possible to restore messages to an iPhone when they're backed up on a computer? I'd like to restore a series of deleted messages (a whole conversation) to an iPhone. The snag is - they're backed up on my computer but not on iCloud.

Is that possible at all?
You can restore the whole backup to your iPhone, which will obliterate/replace all of what is on your iPhone now with the contents of that backup. Unfortunately the restore is an all-or-nothing thing, and replaces all of what is on the iPhone now. Also a big caveat that if you use iCloud Messages syncing, then iCloud will just tell the iPhone to "re-delete" those messages from the phone again shortly after you restore it, since their deleted state is tracked in iCloud. (You can see if you use Messages syncing by going to Settings app > your name > iCloud, and see if the Messages switch is turned on.

Another alternative is to read the messages to your computer out of the backup where they're not deleted. This will at least give you access to the messages on your computer but not restored to the iPhone. There's a few programs to do this. One that I am a developer for is called Decipher TextMessage https://deciphertools.com/decipher-textmessage.html 

Honesty disclaimer: I am a developer/owner of the company that I just linked to, so I stand to benefit, financially or otherwise, from my link. ;)