Can I look at/recover my old texts on the computer

The only way to get deleted texts back is if you have a backup from before you deleted the texts. And you can't just get the deleted texts. You'd have to restore your phone from that backup, which will overwrite anything since you made the backup. If you have backed up since you deleted the texts, they are gone.

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

As a last resort, you can always ask the recipient of the texts to forward them back to you.

if you backup to iTunes on the computer

http://www.imactools.com/iphonebackupviewer/

more

extract data from iphone backup

if one backup to iCloud then one likely need to restore the backup on an iOS device

That's correct. Backups are all or nothing and you can not just look at text messages. It's something we've Apple may be working on for a future update, but at this time this is how it works.