If there is no operating system on your phone it might be an issue to back it up properly.
I do not own a FP1 but for my phone it works that you remove the battery and still use the phone while it is plugged in.
If your phone works without a battery being just plugged in, you can read yourself [into this][1]. That might help you getting your data from the phone. MyPhoneExplorer might not work for your case if you have no working OS. Your phone is then “soft bricked”. The phone is called “hard bricked” if (for you) it is not possible anymore to install an OS. Maybe [this][2] will help you installing a working OS after you backed things up. However be reminded that trying to mess with the OS of your phone might void warranty. If you still have warranty, send it in after you backed things up.
PS: Did you read this: Fairphone does not boot anymore
[1]: http://www.yodot.com/android-data-recovery/from-bricked-android.html
[2]: http://´http://forum.xda-developers.com/showthread.php?t=1560117