Is it possible to backup my phone, and restore it to my computer to run under and android emulator? If so, how would I go about doing that?
Theoretically, it should be possible. In order to do it, you need a custom recovery in your phone to perform the backup and a custom recovery, too, to perform the replacement in the emulator.
TWRP has a custom recovery for the Android emulator and it's currently under TWRP's support. If you have the same recovery in your phone, it could be an option to perform the backup (make sure you're backing up only the data and not everything else, system images between your device and the emulator are probably different).
Although seems like an impossible task, by playing with the TWRP settings you could get your phone's backup to work on your emulator. You just need to handle the inconsistencies between the two of them.