If you were syncing your Notes with your iCloud account (not backing up to iCloud, I mean using the Notes aspect of your iCloud account) you can log into your iCloud account at www.icloud.com from a computer (not iOS device) and access your notes that way. If you were syncing your Notes to Yahoo or Google, you may have similar luck logging into those accounts.
If you weren't syncing Notes to any e-mail account, then they only lived on your device (but are part of your backup, assuming your were backing up your iPhone). In that case you can get them by installing your backup onto a new iPhone (assuming you go that route) or possibly using a 3rd party application to read your backup file to extract the notes (though I've never done that, and not entirely sure it is possible).