Sorry to hear about the issues you have been having with your iPhone 6! There are several things that you can try to fix this issue, however they generally involve restoring your device, which will mean you lose your data. Before you try this plug your device into iTunes. Does your iPhone appear in iTunes when plugged in? If it does, try and attempt a backup. This means that your data will be stored on the computer and you should be able to put it back on when the phone is fixed.
If this doesn't work then it is unlikely that your data will be recoverable. You could try one of the many data recovery programs available on the internet, but these often have limited success. Is your iPhone set to back up to iCloud? It might be without you even knowing. You can check this by follow this Apple support document. It also shows you how to see when your last backup was on iTunes.
If you wish to continue and restore your device please follow the below guide to restore your iPhone:
Restore Your iPhone
If this doesn't work you can try a DFU mode restore, following the instructions below:
DFU Mode Restore
Another user may have an idea on how you can save your data and fix your phone at the same time, but I cannot think of how.
All the best for your repair!