My computer was formatted but I need to access my old content on my itunes backups. Is there any way I can acquire these iTunes backup files ever again?

You reformatted the drive in your computer. The backup files you want to access were overwritten when you re-formatted the drive. They are gone. There may be data recovery services that can get them back. Search google if you want. This has absolutely nothing to do with using an iPhone at this point. You deleted files on your computer...

If you formatted your hard drive, all data was erased. Unless you made a backup before you formatted, your data is gone.