How can I recover old text messages ?

If you use the same Apple ID, that backup has likely been overwritten many times in the last year. If it was just a few weeks ago, that would be possible, but from a year ago, likely long gone.

Read this if you use iCloud Bakup:

If you use iCloud Backup, you won't find a list of your backups on iCloud.com. Here's how to find your iCloud backups on your iOS device, Mac, or PC.

On your iPhone, iPad, or iPod touch:

• Using iOS 11, go to Settings > [your name] > iCloud > Manage Storage > Backup.
• Using iOS 10.3, go to Settings > [your name] > iCloud. Tap the graph that shows your iCloud usage, then tap Manage Storage. 
• Using iOS 10.2 or earlier, go to Settings > iCloud > Storage > Manage Storage.

Tap a backup to see more details about it, choose which information to back up, or delete the backup.

When were they deleted? If you have a backup from before the messages were deleted, you could restore from that backup. If you don't and the messages were deleted, there is no way to get them back.

No. If the backup is housed on a computer, it's on the computer and nowhere else.